Table of contents xi

. The point is that pseudoeffectivity of the tangent, or reflexivized cotangent sheaf of a variety detects an abelian factor in its singular Beauville-Bogomolov decomposition.

Theorem 1.1. Let X be a normal projective variety with klt singularities and numerically trivial K X . If its tangent or reflexivized cotangent sheaf is pseudoeffective, then there is a quasiétale finite cover X → X such that q( X) ̸ = 0. Equivalently, the singular Beauville-Bogomolov decomposition of X has an abelian factor of positive dimension.

In particular, if X is a singular Calabi-Yau or IHS variety in the sense of Definition 5.2, then neither T X nor its dual Ω

[1]

X is pseudoeffective. This result generalizes [86, Theorem 1.6], which makes a technical hypothesis of smoothness in codimension 2. The proof uses the singular Beauville-Bogomolov decomposition theorem and the following result.

Theorem 1.2. Let X be a normal projective variety with klt singularities of dimension n, H a Q-Cartier ample divisor on X. Consider E a reflexive sheaf on X such that c 1 (E) • H n-1 = 0, the sheaves E and S [l] E, for some l ≥ 6, are H-stable, and E is pseudoeffective. Then c 1 (E) 2 = c 2 (E) = 0.

Moreover, there is a quasiétale finite Galois covering ν : X → X, such that ν [ * ] E is locally-free, has a numerically trivial determinant, and is Gal( X/X)-equivariantly flat on X, i.e., comes from a Gal( X/X)-equivariant representation of π 1 ( X). In particular, ν [ * ] E is numerically flat, and

, and that is a key ingredient in the proof of the singular Beauville-Bogomolov decomposition theorem too.

As for its proof, first note that positivity of a sheaf is not preserved by birational modifications, hence we could not simply terminalize X into X and use the mentioned theorems on X. Hence, the strategy is to first reduce the generalization of Theorem 1.2, Theorem 3.3, to a statement on a klt surface S. By one of the standard construction for orbifold Chern classes, we then construct from the positive reflexive sheaf E on the surface S with quotient singularities a locally free sheaf Ê on a finite Galois cover Ŝ of S: the Chern classes of Ê correspond to the orbifold Chern classes of E. We then play back-and-forth between different notions of positivity for E and Ê.

.

RÉSUMÉ

Cette thèse participe à la description de certaines variétés complexes projectives à diviseur anticanonique numériquement effectif (nef).

Dans la première partie, nous montrons que les faisceaux tangent et cotangent réflexivisé d'une variété normale projective de Calabi-Yau ou irréductible holomorphe symplectique à singularités canoniques ne sont pas pseudoeffectifs, ce qui généralise des résultats de Höring et Peternell en retranchant une hypothèse de lissité en codimension 2. La positivité de la seconde classe de Chern orbifold joue un rôle important dans la preuve, notamment dans un théorème technique faisant le lien entre la pseudoeffectivité d'un faisceau réflexif suffisamment stable de déterminant trivial et l'annulation de sa seconde classe de Chern orbifold. Nous présentons également des exemples de variétés de Calabi-Yau de petite dimension ayant des singularités en codimension 2.

Dans la deuxième partie, nous exposons plusieurs résultats liés à la classification des quotients de variétés abéliennes par des groupes finis agissant librement en codimension 2 qui admettent une variété de Calabi-Yau comme résolution. Il est équivalent de classifier les variétés de Calabi-Yau admettant une annulation partielle de la seconde classe de Chern. Tandis qu'Oguiso construit deux exemples en dimension 3, nous prouvons qu'il n'y en a pas en dimension 4. Nous montrons également qu'à dimension fixée et à isogénie près, il y a seulement deux variétés abéliennes susceptibles d'admettre de tels quotients, à savoir E -1+i √ . Quant au groupe fini agissant, nous montrons qu'il est engendré par ses éléments admettant des points fixes, et nous classifions ses sous-groupes de la forme PStab(a), fixant un point commun a de la variété abélienne étudiée : ces sous-groupes sont des 3-groupes ou des 7-groupes abéliens élémentaires. Finalement, nos résultats impliquent qu'aucun quotient de variété abélienne par un groupe agissant librement en codimension 3 n'admet de résolution crépante simplement connexe.

Le but de la troisième partie est d'établir la conjecture du cone pour les paires de Schoen (une terminologie que nous introduirons), généralisant l'article de Grassi et Morrison sur les variétés de Calabi-Yau de dimension 3 introduites par Schoen. Pour prouver cette conjecture dans ce cas particulier, nous décrivons complètement le cone nef des variétés de Schoen, en utilisant leur description en tant que produits fibrés au dessus de P 1 . Ce travail est une collaboration avec Hsueh-Yung Lin et Long Wang.

Dans la quatrième partie, nous prouvons qu'une variété X projective lisse de dimension n ≥ 4, respectivement n ≥ 5, dont la troisième, respectivement quatrième, puissance extérieure du fibré tangent est strictement nef est une variété de Fano. Nous classifions également une telle variété X sous l'hypothèse additionnelle ρ(X) ̸ = 1. Enfin, nous prouvons que si la (n -1)-ième puissance extérieure du fibré tangent est nef et X est rationnellement connexe, alors X est une variété de Fano. v Mots clés : variétés de Calabi-Yau, variétés irréductibles holomorphes symplectiques, variétés abéliennes, positivité du faisceau (co)tangent, classes de Chern, correspondence de McKay, résolution crépante, cône nef, conjecture du cône de Kawamata-Morrison, variétés rationnellement connexes, variétés de Fano.

ABSTRACT

This thesis contributes to the description of some complex projective varieties with nef anticanonical divisor.

In the first part, we prove that the tangent and the reflexivized cotangent sheaves of any normal projective Calabi-Yau or irreducible holomorphic symplectic variety with canonical singularities are not pseudoeffective, generalizing results of Höring and Peternell by removing an assumption of smoothness in codimension 2. Positivity of the second orbifold Chern class plays a key role in the proof, namely in a technical theorem relating pseudoeffectivity of a sufficiently stable reflexive sheaf with trivial determinant to the vanishing of its orbifold second Chern class. We also provide examples of Calabi-Yau varieties of small dimension with singularities in codimension 2.

In the second part, we present many results toward a classification of those quotients of an abelian variety by a finite group acting freely in codimension 2 that admit a Calabi-Yau resolution. This is equivalent to classifying Calabi-Yau manifolds with a partial vanishing of the second Chern class. While Oguiso constructed two examples in dimension 3, we show that there are none in dimension 4. We also show that, up to isogeny, there are only two abelian varieties admitting such finite quotients in each dimension: E -1+i √ . As for the finite group acting, we show that it is generated by its elements admitting fixed points, and classify its subgroups Pstab(a) that have a common fixed point a: they are elementary abelian 3-groups or 7-groups. Finally, our results imply that no quotient of an abelian variety by a finite group acting freely in codimension 3 admits a simply-connected crepant resolution.

The goal of the third part is to establish the Cone Conjecture for so-called Schoen pairs, generalizing the work by Grassi and Morrison on the Calabi-Yau threefolds constructed by Schoen. In order to prove it, we completely describe the nef cone of Schoen varieties, using their description as fiber products over P 1 . This is joint work with Hsueh-Yung Lin and Long Wang.

In the fourth part, we prove that a smooth projective variety X of dimension n ≥ 4, respectively n ≥ 5, with strictly nef third, respectively fourth, exterior power of the tangent bundle is a Fano variety. We also classify X under the assumption that ρ(X) ̸ = 1. Finally, we prove that if the (n -1)-th exterior power of the tangent bundle is nef and X is rationally connected, then X is a Fano variety.

Keywords: Calabi-Yau varieties, hyperkähler varieties, abelian varieties, positivity of the (co)tangent sheaf, Chern classes, McKay correspondence, crepant resolution, nef cone, Kawamata-Morrison Cone Conjecture, rationally connected varieties, Fano varieties. vii viii Of all the things I know, I cannot count how many I learned from talking to people. Therefore, as my graduate years draw to an end, I have to pay my respects and gratitude to all the teachers, colleagues, and friends who have enlightened my journey toward mathematics, and complex algebraic geometry.

My most sincere thanks go to my thesis supervisor Andreas Höring. I have learned a lot from him since he accepted to supervise my Master's thesis about four years ago: be it about which geometric intuition one should have, how to write mathematics properly, how long to focus on a strenuous problem, or which conferences to go to, his advice and encouragements have always felt appropriate and honest, and I shall miss them soon. Our regular discussions have meant so much to my determination to always learn more and investigate further, and they have only made my enthusiasm for complex algebraic geometry grow. I am in his debt for his extraordinary patience and commitment.

As the referees of my thesis, Daniel Greb and Claire Voisin have provided me with their helpful and benevolent comments. This version of the manuscript has been improved by their suggestions, and I am grateful for their careful work. I also want to thank Arnaud Beauville, Junyan Cao, Enrica Floris, and Christoph Sorger, who honored me by accepting to be on my jury. Over the last three years, my comité de suivi has been patiently listening to my progresses and problems, always offering appropriate advice and encouragement. They have always made sure that my working conditions were good and that my projects stayed on track. Thanks to Indira Chatterji, Enrica Floris, and Christophe Ritzenthaler!

The JSPS Summer Program offered me the great opportunity to go on a research visit at the University of Tokyo last summer. I want to thank Keiji Oguiso for hosting me there and allowing me to participate in the Japanese mathematical life. I thank my collaborators Hsueh-Yung Lin and Long Wang for the fruitful discussions before and during my stay; I am glad that they accepted that our joint work be part of this thesis. I also thank Akira Ishii and Sho Tanimoto for their warm welcome at Nagoya University, and Yukari Ito for her invitation to visit Kavli IPMU. Finally, thanks to the secretary in charge of the JSPS at the University of Tokyo, Ms. Aki Ikeda, for guiding me through the administrative duties bound to my visit.

On a regular basis, I have had the opportunity to interact with a large pool of talented and open-minded colleagues. So many discussions changed my perspective on certain parts of algebraic geometry, sharpened my understanding of the broader picture, and made me excited about learning new things and starting new projects. Let me credit them, in alphabetical order: Anna Bot, Fabio Bernasconi, Benoît Cadorel, Benoît Claudon, Tien-Cuong Dinh, Stéphane Druel, Stefano Filipazzi, Salvatore Flocix cari, Henri Guénancia, Yoshinori Gongyo, Stéphane Lamy, Christian Lehn, Frank Loray, Giancarlo Lucchini-Arteche, Shin-ichi Matsumura, Martina Monti, Anne Moreau, Zsolt Patakfalvi, Julia Schneider, Saverio Secci, Robert Śmiech, Roberto Svaldi, Gianluca Pacienza, Xun Yu. I also thank all the great anonymous organizers who worked to make these encounters possible, helping the complex algebraic geometry research network recover and grow back after the pandemic.

A lot of people helped me navigate graduate school in non-mathematical ways. I thank the former and current heads of the doctoral school, Élizabeth Taffin de Givenchy and Giovana Tissoni, and the secretary of the ATG team, Clara Salaun, for patiently answering my poorly informed administrative questions. I also thank the outreach team, in particular Marc and Valérie, for offering a welcoming environment for experimenting ways of explaining mathematics to a broad audience.

These acknowledgements are an occasion to credit some of the people whose influence has been a driving force in my young adult life. From my high-school years, I should thank Jean-Marc Bryard and Éric Oudin, who taught me so much about how to read and write an essay, be it in philosophy or history, in French or in German. As I write these words, in yet another topic and language, I remember their teachings. I am also in debt to my teachers Alain Troesch and Yves Duval, notably as they lit my spark for group theory. After a short research internship supervised by Misha Verbitsky, I got better acquainted with complex geometry through the lectures by Ilia Itenberg, Julien Marché, Antoine Ducros, and Olivier Debarre. I am grateful to all these people for contributing to my mathematical path.

The atmosphere among graduate students at the lab in Nice is cheerful and inclusive, and much of that is due to Alex, Angel, Antoine C., Antoine E., Benjamin, Charbel, Chloé, Dahmane, Gustave, Jérémie, Lamine, Matthieu, M'hammed, Nadine, Sumithra, Thanos, Thibault, Victor, Yash, Zakaria: Many thanks to you all! Special thanks go to Najwa for being kind, funny and positively stubborn. A few colleagues, although not quite graduate students anymore, have been just as cheerful and inclusive, like Didier and his tangerine trees, Jérémy and his moka pot that saves the lives of baby dolphins, Joachim, Michele, Stéphanie, Vincent. So long, everyone, and I wish you all the best. I had not suspected that becoming a graduate student would make me part of an academic family. My younger sibling Bruno, and my older siblings Yanis, Zhining and Zhixin, have been colleagues, friends, and so much more. I thank them.

I am probably old enough now where it is slightly embarrassing for me to admit having a role model. But let me embrace the awkwardness for once, as I express my intellectual debt to the early movies of Sylvester Stallone. Be it Rambo, Rocky or the person behind them, they taught me about tough mindedness, humanity, and the balance between them; they inspired me; they contributed to my growth as a mathematician, and as a person.

To my brother, sister and parents, to my grandma, to my fellow brass band people, to my friends here in Nice, there in Paris and its suburbs, and over there in Toulouse, goes my profound gratitude, for their constant material and emotional support, their timely presence whenever I needed something different, and their understanding whenever seminars, conferences, research visits, or the urge to finish a project overwhelmed me. And to Wesley, my love for whom instills every word I write: Thank you.

x CHAPTER 1

INTRODUCTION

1.1 Introduction. Ever since the Greek antiquity, correspondences between geometric shapes and equations have been at the root of geometry, algebra, and arithmetics at once. Diophantine arithmetics considers integral, or rational solutions to polynomial equations, while geometry and algebra accept real, or complex solutions. A complex projective variety X is the set of solutions to a system of irreducible complex polynomial equations in the complex projective space P n . Smoothness of the variety X is equivalent to an algebraic property of the corresponding system of polynomial equations, as says the Jacobi criterion.

Smooth complex projective varieties are complex manifolds: as such, they come naturally equipped with a tangent bundle. Positivity properties of this tangent bundle recover information about the initial variety. For instance, if the tangent bundle is ample, the initial variety is a projective space, whereas if the tangent bundle is numerically flat, the initial variety is an étale quotient of an abelian variety. However, the tangent bundle of a given variety is rarely fully understood. What we might know, given a variety, is some invariants attached to its tangent bundle, the Chern classes.

Chern classes are a useful tool in many ways. Alike Stiefel-Whitney classes, they can provide obstructions to embedding certain varieties into low dimensional projective spaces, as in [START_REF] Van De Ven | On the embedding of abelian varieties in projective spaces[END_REF], where it is proven that for d ≥ 3, a d-dimensional abelian variety cannot be embedded in P 2d . Chern classes (notably of the (co)tangent bundle) satisfy inequalities, as we will illustrate in Section 2.7. Some of the equality cases in these inequalities are characterized as peculiar geometric situations as well. These equality cases are noteworthy, as they allow to recover information on the geometry of X from purely numerical invariants.

This thesis, as its title suggests, deals with positivity of vector bundles, mirrored into positivity of its Chern classes. The goal is fourfold: in Part I, we prove that a slightly positive reflexive sheaf with vanishing first Chern class has vanishing second Chern class. Still in Part I, but also in Part II, we use information about a second Chern class (a positivity condition in Part I, and a vanishing in Part II) to derive information about a variety (about the positivity of its tangent bundle in Part I, about the variety itself in Part II). In Part III, we change from varieties of trivial first Chern class to varieties with non-negative first Chern class. A conjecture, the Kawamata-Morrison Cone Conjecture, describes the cone of non-negative divisors on such varieties. We prove it in a particular case. Finally, Part IV is about proving that some varieties with non-negative first Chern class, under a positivity assumption on some vector bundle canonically associated to them, actually have positive first Chern class.

Let us sketch our work in more detail.

Part II.

In Part II, we focus on smooth Calabi-Yau varieties X admitting a nef and big divisor D with c 2 (X) • D n-2 = 0. Equivalently, by Theorem 2.57, these Calabi-Yau varieties are resolutions of quotients of abelian varieties by a finite group acting freely in codimension 2. In dimension 3, these are precisely the Calabi-Yau varieties of type III 0 in the classification of Calabi-Yau algebraic fiber spaces [START_REF] Oguiso | On algebraic fiber space structures on a Calabi-Yau 3-fold[END_REF], that are determined by Oguiso in [START_REF] Oguiso | On the complete classification of Calabi-Yau threefolds of type III 0[END_REF]: there are exactly two rigid instances of them. Describing such varieties plays an important role in the classification of contractions of Calabi-Yau manifolds. This classification, and in particular [START_REF] Oguiso | On the complete classification of Calabi-Yau threefolds of type III 0[END_REF], is for instance used in [START_REF] Oguiso | Calabi-Yau threefolds with positive second Chern class[END_REF] to prove that in a general Calabi-Yau hypersurface X in a smooth Fano fourfold, every nef divisor D satisfies c 2 (X) • D > 0.

We pursue the same purpose in higher dimension, and achieve it in dimension 4: there are no simply-connected crepant resolutions of quotients of abelian fourfolds by finite groups acting freely in codimension 2. We obtain partial results indicating that examples should be sparse in arbitrary dimension too, proving, e.g., the following theorem, where we denote by j the first primitive third root of unit and by u 7 the quadratic integer -1+i √ 7 2

.

Theorem 1.3. Let A be an abelian variety of dimension n and G be a finite group acting freely in codimension 2 on A. If A/G has a crepant resolution that is a Calabi-Yau manifold, then

(1) A is isogenous to E j n or to E u 7 n , and G is generated by its elements that admit fixed points.

(2) For every translated abelian subvariety W ⊂ A, there is k ∈ N such that the pointwise stabilizer

PStab(W ) := {g ∈ G | ∀w ∈ W, g(w) = w} is isomorphic to Z 3 k if A is isogenous to E j n , or to Z 7 k if A is isogenous to E u 7 n .
(3) For every translated abelian subvariety W ⊂ A, if PStab(W ) is isomorphic to

• Z 3 k , then there are k generators of it such that their matrices are similar to diag(1 n-3 , j, j, j), and the j-eigenspaces of these matrices are in direct sum.

• Z 7 k , then there are k generators of it such that their matrices are similar to diag(1 n-3 , ζ 7 , ζ 7 2 , ζ 7 4 ), and all eigenspaces of these matrices with eigenvalues other than 1 are in direct sum.

Finally, we prove that there are no simply-connected crepant resolutions of quotients of abelian varieties by finite groups acting freely in codimension 3.

On the way, we also prove a result in the spirit of a conjecture by Ito and Reid in McKay correspondence [START_REF] Ito | The McKay correspondence for finite subgroups of SL(3, C). In Higher-dimensional complex varieties[END_REF], and find a counterexample to the conjecture itself.

Conjecture 1.4. Let G < GL n (C). If the quotient C n /G admits a crepant resolution, then every maximal cyclic subgroup of G is generated by a junior element.

As a counterexample, we propose a representation of the group SL 2 (F 3 ) in GL 6 (C), with a maximal cyclic subgroup Z 4 that is not generated by a junior element, but such that the quotient C 6 /SL 2 (F 3 ) admits a crepant resolution (this last part being checked with help of the software Macaulay2 [START_REF] Grayson | 2, a software system for research in algebraic geometry[END_REF]). We correct the conjecture with a milder result.

Proposition 1.5. Let G < GL n (C). If the quotient C n /G admits a crepant resolution, then G is generated by junior elements.

This part relies on the numerous properties of automorphisms of small dimensional abelian varieties [START_REF] Birkenhake | Complex abelian varieties[END_REF]Chapter 13], and on heavy computational group theory. We make good use of the SmallGrp package of the software GAP [START_REF]GAP -Groups, Algorithms, and Programming[END_REF]. The results of this part are the subject of a preprint [START_REF] Gachet | Finite quotients of an abelian variety with a Calabi-Yau resolution[END_REF].

1.4 Part III. In Part III, which is joint work with Hsueh-Yung Lin and Long Wang, we describe the nef cone of some Calabi-Yau pairs whose underlying varieties are obtained by fiber products of certain rationally-connected manifolds fibred over P 1 . This generalizes a construction of Calabi-Yau threefolds from rational elliptic surfaces due to Schoen [START_REF] Schoen | On fiber products of rational elliptic surfaces with section[END_REF]. Our description of the nef cone enables us to prove the Kawamata-Morrison Cone Conjecture for our examples, i.e., that the action of a certain subgroup Aut(X, ∆) of the automorphism group on the nef effective cone of X admits a rational polyhedral fundamental domain. The Kawamata-Morrison Cone Conjecture is a longstanding conjecture for klt pairs (and notably varieties) with trivial canonical class, and although it was proven in full generality in dimension 2 [START_REF] Totaro | The cone conjecture for Calabi-Yau pairs in dimension 2[END_REF], it is still widely open in dimension 3. We refer the curious reader to [START_REF] Lazić | The Morrison-Kawamata cone conjecture and abundance on Ricci flat manifolds[END_REF] for a survey of its history and implications, as well as to Chapter 17.

The decomposition of the nef cone is proven by ad hoc methods, relying heavily on the fact that we consider fiber products over a curve (see Examples 19.5,19.6). The result is the following. Theorem 1.6. For i = 1, 2, let ϕ i : W i → C be a surjective morphism with connected fibers from a smooth projective variety to a smooth projective curve. Assume that

• the variety W = W 1 × C W 2 is smooth; • it holds p * 1 N 1 (W 1 ) R + p * 2 N 1 (W 2 ) R = N 1 (W ) R
, where p i denotes the projection from W onto W i .

Then p * 1 Nef(W 1 ) + p * 2 Nef(W 2 ) = Nef(W ). As a consequence, p * 1 Amp(W 1 ) + p * 2 Amp(W 2 ) = Amp(W ).

Grassi and Morrison [START_REF] Grassi | Automorphisms and the Kähler cone of certain Calabi-Yau manifolds[END_REF]Proposition 3.1] had already proven such a nef cone decomposition for the fiber product of two rational elliptic surfaces with their natural fibrations to P 1 .

The existence of the rational polyhedral domain then follows, using a result by Totaro [START_REF] Totaro | Hilbert's 14th problem over finite fields and a conjecture on the cone of curves[END_REF]Theorem 8.2] for rational elliptic surfaces, a result by Kollár [START_REF] Borcea | Homogeneous vector bundles and families of Calabi-Yau threefolds. II. In Several Complex Variables and Complex Geometry[END_REF] for ample hypersurfaces in Fano manifolds, a result by Looijenga [START_REF] Looijenga | Discrete automorphism groups of convex cones of finite type[END_REF] on the existence of a fundamental domain for a linear action on a cone, and by carefully choosing subgroups of the automorphism group of the factors of the fiber product, that are large enough to both act extensively on the nef cone, and small enough that their product embeds in Aut(X, ∆). This strategy allows to bypass any decomposition result at the level of automorphism groups, while such a decomposition [START_REF] Namikawa | On the birational structure of certain Calabi-Yau threefolds[END_REF]Corollary 2.3] was crucial to Grassi and Morrison's proof [START_REF] Grassi | Automorphisms and the Kähler cone of certain Calabi-Yau manifolds[END_REF]. These results should be available on the arXiv soon.

1.5 Part IV. In Part IV, we investigate some varieties with nef first Chern class, namely smooth varieties X such that an exterior power of the tangent bundle r T X is strictly nef. Definition 1.7. A line bundle L on a normal projective variety X is strictly nef if, for every curve C in X, the intersection number L • C is positive. A vector bundle E is strictly nef if, on its associated projectivized bundle P(E), the tautological line bundle is strictly nef.

For r = dim(X), a conjecture by Campana and Peternell [START_REF] Campana | Projective manifolds whose tangent bundles are numerically effective[END_REF]Problem 11.4] predicts that smooth varieties with strictly nef anticanonical divisor are Fano varieties. It is quite a surprising conjecture, as in general, a strictly nef divisor has no reason to be ample [START_REF] Hartshorne | Ample subvarietes of algebraic varieties[END_REF]Chapter 1,Example 10.6], but if true, it would provide a simplified version of Nakai-Moishezon criterion for ampleness of -K X . Until now, this conjecture is however only known in dimension up to 3, by the work of Maeda [START_REF] Maeda | A criterion for a smooth surface to be Del Pezzo[END_REF] and Serrano [START_REF] Serrano | Strictly nef divisors and Fano threefolds[END_REF].

We investigate a similar question for different values of r.

Question 1.8. Let X be a smooth projective variety of dimension n. Let 1 ≤ r ≤ n be an integer, and assume that r T X is strictly nef. Is X a Fano manifold?

In the case r = 1, Li, Ou and Yang [START_REF] Li | On projective varieties with strictly nef tangent bundles[END_REF] prove that X must be a projective space. In the case r = 2 and dim(X) ≥ 3, they prove that X must be a projective space or a quadric hypersurface, hence a Fano manifold. We also prove that X is Fano when r = 3 and dim(X) ≥ 4, when r = 4 and dim(X) ≥ 5, and when r = dim(X) -1. In the first two cases, we present a classification of those varieties X which additionally satisfy ρ(X) ̸ = 1, based on the theory of Mori contractions of large length. Theorem 1.9. Let X be a smooth projective rationally connected variety of dimension n ≥ 4 such that for each rational curve C in X, we have -K X • C ≥ n -1. Then either X ≃ P 2 × P 2 , or X is a Fano variety of Picard rank ρ(X) = 1.

Corollary 1.10. Let X be a smooth projective variety of dimension at least 4 such that the vector bundle 3 T X is strictly nef. Then either X ≃ P 2 × P 2 , or X is a Fano variety of Picard rank ρ(X) = 1.

Theorem 1.11. Let X be a smooth projective rationally connected variety of dimension n ≥ 6 such that for each rational curve C in X, we have -K X • C ≥ n -2. Then either X is isomorphic to P 3 × P 3 or X is a Fano variety of Picard rank ρ(X) = 1.

Corollary 1.12. Let X be a smooth projective variety of dimension at least 5 such that the vector bundle 4 T X is strictly nef. Then either X is isomorphic to one of the following Fano varieties P 2 × Q 3 ; P 2 × P 3 ; P(T P 3 ); Bl ℓ (P 5 ) = P(O P 3 ⊕ O P 3 ⊕ O P 3 (1)); P 3 × P 3 or X is a Fano variety of Picard rank ρ(X) = 1. Theorem 1.13. Let X be a smooth projective rationally connected variety of dimension n such that the vector bundle n-1 T X is nef. Then X is a Fano variety.

The proof of this last theorem relies on Chern classes inequalities à la Demailly, Peternell, Schneider [START_REF] Demailly | Compact complex manifolds with numerically effective tangent bundles[END_REF], and on the Hirzebruch-Riemann-Roch formula. Our result and its proof are reminiscent of the fact known by [START_REF] Demailly | Compact complex manifolds with numerically effective tangent bundles[END_REF], that a smooth rationally connected variety X such that T X is nef is a Fano variety. Note that, building on this theorem, [START_REF] Watanabe | Positivity of the exterior power of the tangent bundles[END_REF]Proposition 1.4] very recently gave an affirmative answer to Question 1.8 in general. This work appears in the author's preprint [START_REF] Gachet | Positivity of higher exterior powers of the tangent bundle[END_REF].

Future directions.

There still are many mysteries regarding positivity of Chern classes. For instance, it is still not known if the second Chern class of a variety with trivial first Chern class is always pseudoeffective, in the sense that it lies in the closure of the effective cone of codimension 2 cycles. In dimension 3, the dual of the cone of pseudoeffetive 1-cycles being the nef cone, the Inequality 2.51 asserts this pseudoeffectivity. However, in higher dimension, the closure of the cone generated by complete intersections of ample divisors is no longer dual to the pseudoeffective cone, and thus the pseudoeffectivity of c 2 (X) is not implied by Inequality 2.51. On pathologies of the nef cone and the pseudoeffective cone for higher codimension cycles, see [START_REF] Debarre | Pseudoeffective and nef classes on abelian varieties[END_REF] and [START_REF] Fulger | Positive cones of dual cycle classes[END_REF]. By work of Ottem [START_REF] Ottem | Nef cycles on some hyperkähler fourfolds[END_REF], it is known that on the variety of lines of a cubic fourfold (which is an irreducible holomorphic symplectic fourfold), the second Chern class is not big, i.e., not in the interior of the pseudoeffective cone. To our knowledge, more is not known.

Closer to our work, note that the classification of the finite quotients A/G that are smooth in codimension 2 and admit a Calabi-Yau resolution in Part II would need to be extended to higher dimensions. Under the additional assumption that the group G is abelian, Theorem 7.6 and the results of Section 12 prove that, if A is an abelian variety of dimension n and G is a finite abelian group acting freely in codimension 2 on A such that A/G admits a Calabi-Yau resolution X, then n = 3 and X is X 3 or X 7 . Also note that G is abelian if and only if any two junior elements g, h of G commute, which by our results can be checked via their matrices acting on a vector space V of dimension 3, 4, 5, or 6. Standard finite group theory allows us to explicitly bound the order of ⟨g, h⟩ depending on this dimension and the isogeny type of A. If the dimension is 3 or 4, the bounds are reasonable enough to launch a computer-assisted search through all possible abstract groups ⟨g, h⟩. Among these, the only groups which, in a faithful 3 or 4-dimensional representation, are generated by two junior elements of the same type, are Z 3 , Z 7 , and the finite simple group SL 3 (F 2 ) of order 168. But a geometric argument on fixed loci excludes SL 3 (F 2 ), whence the wished contradiction. This reproves the classification of [START_REF] Oguiso | On the complete classification of Calabi-Yau threefolds of type III 0[END_REF] in dimension 3, and settles Theorem 7.5. When V has dimension 5 or 6, we could also bound the order of ⟨g, h⟩ explicitly, but the bounds obtained in this way are too large for the SmallGrp library. In order to rpove that G is abelian, one needs to build a reasonably smaller finite list of possibilities for the abstract group ⟨g, h⟩, and to rule out those potential groups in the list other than Z 3 , Z 7 , Z 3 × Z 3 , and Z 7 × Z 7 . This is our work in progress in dimension 5. There are open questions about the known quotients of dimension 3 too. For instance, the discussion of rational curves on quotients of abelian varieties is a topic of long-lasting interest [START_REF] Kollár | Quotients of Calabi-Yau varieties[END_REF][START_REF] Im | Rational curves on quotients of abelian varieties by finite groups[END_REF], connected to the topic of the gonality of curves on abelian varieties [START_REF] Pirola | Curves on generic Kummer varieties[END_REF][START_REF] Voisin | Chow ring and gonality of general abelian varieties[END_REF]. It would be interesting to prove that the number of rational curves on E -1+i √ 7 2

3

/Z 7 is finite, or that the uniruled divisors in (E j ) 3 /⟨diag(j, j, j)⟩ are exactly those induced by inclusions of the two-dimensional factor (as (E j ) 2 /⟨diag(j, j)⟩ is a rational surface). Such considerations might lead to a proof of the Cone Conjecture for Oguiso's Calabi-Yau threefolds X 7 and X 3 . In fact, these two Calabi-Yau threefolds present a dichotomy: X 7 has a finite automorphism group, and the quotient

E -1+i √ 7 2

3

/Z 7 has a rational polyhedral nef cone, while X 3 has an infinite automorphism group, and the quotient (E j ) 3 /⟨diag(j, j, j)⟩ has a circular nef cone. However, it is not clear how to describe the nef cones of X 7 and X 3 , as their relative Picard numbers above the quotients they resolve are quite large (21, respectively 27). A first attempt could be to find an alternative prove of the Cone Conjecture for the Kummer surface, which would bypass the use of lattice theory coming along with the realm of K3 surfaces, and the description of its Mori cone by [START_REF] Kovács | The cone of curves of a K3 surface[END_REF].

Another question stemming from Part II is the following: are there sufficient criteria for affine quotients C n /G by finite non-abelian groups G acting freely in codimension 1 to admit a crepant resolution? In the case that they admit a crepant resolution, how many do they have? In the abelian case, toric geometry provides a powerful tool to bring partial answers to these questions, see e.g. [START_REF] Sato | Crepant property of Fujiki-Oka resolutions for Gorenstein abelian quotient singularities[END_REF], a tool which lacks the non-abelian case. Partial results in the non-abelian case include [START_REF] Lehn | A symplectic resolution for the binary tetrahedral group[END_REF][START_REF] Bellamy | Counting resolutions of symplectic quotient singularities[END_REF][START_REF] Donten-Bury | On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32[END_REF][START_REF] Bellamy | All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces[END_REF].

It would also be worthwhile to work on finite quotients of abelian varieties containing some pseudo-reflections. For instance, there is a structure theorem for smooth finite quotients of abelian varieties, due to [START_REF] Auffarth | Smooth quotients of abelian surfaces by finite groups that fix the origin[END_REF][START_REF] Auffarth | Smooth quotients of complex tori by finite groups[END_REF], and a more general structure theorem building finite quotients of abelian varieties from Q-abelian finite quotients of abelian varieties and Q-Fano finite quotients of abelian varieties [START_REF] Shibata | Q-abelian and Q-fano finite quotients of abelian varieties[END_REF]. It seems to us that Q-Fano finite quotients of abelian varieties are worth more investigation, e.g., along the lines of [START_REF] Shibata | Q-abelian and Q-fano finite quotients of abelian varieties[END_REF]Question 5.4]: are they toric?

Finally, Part IV leaves the classification of Fano varieties X with Picard number one and 3 T X or 4 T X strictly nef open. We conjecture that, if X is Fano variety of Picard number one, 3 T X is ample if and only if X is a del Pezzo manifold.

We hope to investigate some of these questions in the future.

CHAPTER 2

PRELIMINARIES

2.1 Notations and conventions. We work over the field of complex numbers C. Varieties (and in particular curves) are always assumed irreducible and reduced. We use the expressions "smooth projective variety" and "projective manifold" interchangeably. We refer to [START_REF] Hartshorne | Algebraic Geometry[END_REF] for scheme theory, [START_REF] Fulton | Intersection theory[END_REF] for intersection theory, [START_REF] Debarre | Higher-dimensional algebraic geometry[END_REF] for birational geometry, in particular Mori theory, [START_REF] Lazarsfeld | Positivity in algebraic geometry[END_REF][START_REF] Lazarsfeld | Positivity in algebraic geometry[END_REF] for positivity notions, [START_REF] Kollár | Rational curves on algebraic varieties[END_REF] for rational curves and their deformations. As regards group theory, we refer to [START_REF] Suzuki | Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF][START_REF] Rotman | An introduction to the theory of groups[END_REF], notably for p-group properties and Sylow theory, and to [START_REF] Huppert | Character theory of finite groups[END_REF] for character theory. When X is a projective variety, we denote by X reg its smooth locus and by X sing its singular locus. Note that, if X is normal, X sing ⊂ X has codimension at least two.

If E is a coherent sheaf on a quasiprojective variety X, we denote by E * its dual sheaf.

Divisors and line bundles. Let X be a normal projective variety. We identify the group Pic(X) = H 1 (X, O * X ) of line bundles up to isomorphism over X with the group of Cartier divisors up to linear equivalence on X. Recall that the exponential short exact sequence induces morphisms of abelian groups:

H 1 (X, O X ) → Pic(X) c 1 → H 2 (X, Z).
We denote by Pic 0 (X) the kernel of c 1 , and by NS(X) the image of c 1 , which we refer to as the Néron-Severi group of X. When k is field, we denote by Pic(X) k the k-vector space Pic(X) ⊗ Z k, and by NS(X) k the k-vector space NS(X) ⊗ Z k. We refer to NS(X) R as the Néron-Severi space of X. It is a theorem [START_REF] Hartshorne | Algebraic Geometry[END_REF]Exercise V.1.8(b)] that the Néron-Severi space of a normal projective variety X is finite-dimensional. Its dimension is called the Picard number of X, and denoted by ρ(X).

Cycles and intersection theory. Let X be a quasiprojective variety. We denote by A k (X), respectively A k (X), the Z-module of cycles of codimension k, respectively of dimension k, up to rational equivalence. A k-cycle is called effective if it can be written as a linear combination of subvarieties of X of dimension k, with positive coefficients. Given two cycles C 1 and C 2 , we write

C 1 = C 2 , or C 1 ∼ C 2 , if they are rationally equivalent.
In [START_REF] Fulton | Intersection theory[END_REF]Chapter 2], an intersection product is defined, that to a k-cycle C and a Cartier divisor D, associates a (k -1)-cycle, and that satisfies the crucial commutation property

D • (D ′ • C) = D ′ • (D • C).
From a k-cycle and k Cartier divisors, we can therefore define an intersection number through the natural group morphism A 0 (X) → Z. C equate. We denote by N 1 (X) R the real vector space generated by classes of 1-cycles, up to numerical equivalence. We denote by N 1 (X) R the real vector space generated by classes of Cartier divisors, up to numerical equivalence. It turns out that

N 1 (X) R ∼ = NS(X) R .
Chern classes on a scheme. In [START_REF] Fulton | Intersection theory[END_REF]Chapter 3] is given a construction of Chern classes c i (E) for a locally free sheaf E on an arbitary scheme X. These Chern classes are defined as operators c i (E) : A * (X) → A * -i (X). One can relate the axiomas given by Grothendieck [START_REF] Grothendieck | La théorie des classes de Chern[END_REF] for Chern classes of locally free sheaves over smooth projective varieties with the properties constituting [START_REF] Fulton | Intersection theory[END_REF]Theorem 3.2].

Chern classes of a variety. When X is a smooth projective variety, we write c i (X) = c i (T X ) for the Chern classes of the tangent bundle of X. When X is a normal projective variety of dimension n, we denote by c 1 (X) = -K X the Weil divisor obtained by taking the Zariski closure of c 1 (X reg ) in X, using the isomorphism

A n-1 (X) ∼ → A n-1 (X reg ) induced by the exact sequence [63, Proposition 1.8].
When X is smooth in codimension 2, e.g., when it is terminal [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]Corollary 5.18], the square of the first Chern class c 1 2 (X) and the second Chern class c 2 (X) are defined similarly, as the codimension 2 cycles obtained by taking the Zarsiki closure of c 2 (X reg ) and c 1 (X reg ) 2 respectively. This process is actually more general, see [START_REF] Druel | A decomposition theorem for singular spaces with trivial canonical class of dimension at most five[END_REF]Definition 2.9]. Note that we will carefully distinguish between c 1 (X) 2 , when X is Gorenstein, and c 1 2 (X), when X is a normal projective variety that is smooth in codimension 2.

Finite morphisms. We will deal with various types of finite maps. Unless otherwise stated, all finite morphisms we speak about are surjective; we may well refer to them as finite coverings.

We say that a finite morphism is quasiétale if it is étale in codimension 1. By Zariski purity of the branch locus, a finite morphism X → Y between normal projective varieties is quasiétale if and only if its branch locus is a closed subscheme of Y sing . Following [START_REF] Greb | Etale fundamental groups of Kawamata-log-terminal spaces, falt sheaves and quotients of abelian varieties[END_REF], we say that a finite morphism of normal varieties X → Y is Galois if it is the quotient map of X by a finite group action. The corresponding group is called the Galois group of the morphism.

Singularities of pairs, and

Calabi-Yau pairs. Throughout this thesis, we will encounter various notions of singularities of the minimal model program (MMP). These singularities are defined for pairs (X, ∆).

Definition 2.1. A pair (X, ∆) is the data of a normal projective variety X and a Q-Weil divisor (or for short, Q-divisor) ∆ that is effective on X such that the Weil divisor

K X + ∆ is Q-Cartier.
Notions that are defined with the canonical bundle for varieties, such as being Fano, or being K-trivial, extend to pairs, as the following definition suggests. Definition 2.2. A pair (X, ∆) is called a Fano pair, respectively a Calabi-Yau pair if -(K X + ∆) is ample, respectively numerically trivial.

To define our MMP-singularities, we first need a few notations. Definition 2.3. Fix a pair (X, ∆). For any proper birational morphism ε : Y → X, there is a formula

m(K Y + ε -1 * ∆) = ε * m(K X + ∆) + a(E i , X, ∆)mE i ,
where the E i are exceptional divisors in the exceptional locus of ε, and m is an integer such that m(K X + ∆) is Cartier. The numbers a(E i , X, ∆) are rational and are called the discrepancies of (X, ∆) with respect to E i . They depend on the divisor E i , but not on the choice of ε or m.

The minimum of the numbers a(E i , X, ∆), running on all exceptional divisors E i of all proper birational morphisms Y → X, is denoted by discrep(X, ∆).

Definition 2.4. If ∆ is an R-divisor, we can write ∆ = i d i D i with d i ∈ R and D i Weil divisors.
The round-down of ∆, which we denote by ⌊∆⌋, is the divisor

i ⌊d i ⌋D i . Definition 2.5. A pair (X, ∆) is called        terminal canonical Kawamata log terminal (klt) log canonical (lc) if discrep(X, ∆)        > 0 ≥ 0 > -1 and ⌊∆⌋ = 0 ≥ -1
A variety X is called terminal, canonical, klt or lc if the pair (X, 0) is.

Example 2.6. A smooth variety is terminal.

Reflexive sheaves.

In this section, we mainly follow the following paper by Hartshorne [START_REF] Hartshorne | Stable reflexive sheaves[END_REF].

Definition 2.7. A coherent sheaf E on a quasiprojective variety X is called reflexive if the canonical sheaf morphism E → E * * from E to its bidual is an isomorphism. Lemma 2.8. [82, Corollary 1.2] If E is a coherent sheaf, then E * is reflexive.
In particular, the tangent sheaf to a quasiprojective variety is a reflexive sheaf. Reflexive sheaves are not so far from locally free sheaves, as the following result specifies. Lemma 2.9. [START_REF] Hartshorne | Stable reflexive sheaves[END_REF]Corollary 1.4] Let E be a reflexive sheaf on a smooth quasiprojective variety X. There is an open set U ⊂ X whose complement has codimension at least three, such that E| U is locally free.

U : E| U → F| U is an isomorphism, then f is an isomorphism.
Restriction to a closed subscheme does not in general preserve reflexivity: think of restricting the tangent sheaf to the cone over a quadric to a smooth line through the vertex. This tangent sheaf is reflexive by Lemma 2.8, but its restriction is not locally free, hence not reflexive by Lemma 2.9. However, the following result holds, and is to come handy in Chapter 4.

Lemma 2.14. Let X be a normal quasiprojective variety, let E be a reflexive sheaf on X, and let H be a globally generated Cartier divisor on X. For a general member

Y ∈ |H|, E| Y is reflexive. Proof. Let U := {(x, Y )|x ∈ Y } ⊂ X × |H|
be the universal family associated to our linear system, with p : U → X and q : U → |H|. the natural projections. As H is globally generated, the fibers of p are hyperplanes in |H|. Hence, the Hilbert polynomial of a fiber is constant, so p is flat by [81, Theorem III.9.9]. So by [START_REF] Hartshorne | Stable reflexive sheaves[END_REF]Proposition 1.8], p * E is reflexive. Now by [START_REF] Dieudonné | Éléments de géométrie algébrique IV. Troisi ème partie[END_REF]Theorem 12

.2.1], for Y ∈ |H| general, p * E| q -1 (Y ) = E| Y is reflexive.
Let E be a reflexive sheaf on a variety X. Recall the reflexivization functor F → F * * enables to perform general algebraic operations in the category of reflexive sheaves.

Notably, we will denote by

• S [m] E the reflexivization of the m-th symmetric power of E (for m ∈ N),

• ν [ * ] E the reflexivization of the pullback of E (for ν : Y → X a surjective morphism).

The reflexive pullback behaves quite differently than the standard pullback, as we will see in Remark 2.25. However, Lemma 2.15. Let p : X → Y be a proper dominant morphism between normal projective varieties, with all fibers of the same dimension. The functor p [ * ] from the category of reflexive sheaves on Y to that of reflexive sheaves on X is left-exact.

Proof. Let 0 → E → F → G → 0 be an exact sequence of reflexive sheaves on Y , and denote by Y sing ⊂ Z ⊂ Y a closed subscheme of codimension at least 2 such that our reflexive sheaves are locally-free on Y \ Z. Reflexive pullback a priori only gives morphisms

p [ * ] E → p [ * ] F → p [ * ] G,
whose composition is zero. By [START_REF]Stacks Project[END_REF]Lemma 31.12.7], the kernel K of the morphism

p [ * ] F → p [ * ] G is reflexive.
There is a natural morphism from p [ * ] E to the kernel K, which restricts to an isomorphism over X \ p -1 (Z) (as pullback is exact for locally free sheaves). As both sheaves are reflexive and p -1 (Z) has codimension 2, they are isomorphic over all X.

We could not produce any counterexample to the right-exactness of this functor. Push-forward is better behaved.

[ * ] E) G ∼ = E.
Proof. As they both are reflexive sheaves, and as there is a canonical map E → (p * p [ * ] E) G , it is enough to show that this map is an isomorphism on the locus where E is locally free, and that is what a local computation shows.

Positivity notions for reflexive sheaves.

We first recall the standard positivity notions for line bundles.

Positivity notions for line bundles.

A line bundle L on a projective variety X is

• globally generated if the sheaf morphism H 0 (X, L) ⊗ O X → L is surjective;

• very ample if it is globally generated and if the induced morphism X → P(H 0 (X, O X (mL)) * ) is an embedding;

• ample if it has a very ample positive multiple;

• strictly nef if for every curve C in X, the intersection number L • C is positive;

• nef if for every curve C in X, the intersection number L • C is non-negative;

• effective if it has a non-zero section;

• pseudoeffective if there is a sequence L n ∈ N 1 (X) R such that for each n, the R-divisor L n is a positive linear combination of effective line bundles, and the sequence (L n ) n∈N converges to L in N 1 (X) R endowed with the natural Euclidean topology.

These definitions motivate the introduction of the convex cones of ample, nef, pseudoeffective, and effective divisors in NS(X) R , Amp(X) ⊂ Nef(X) ⊂ Eff(X) ⊃ Eff(X).

In Part III, we also study the following convex cones: the nef effective cone Nef e (X) := Nef(X) ∩ Eff(X), and the positive nef cone Nef + (X), which is the convex hull of Nef(X) ∩ NS(X) Q in NS(X) R . These two cones play a central role in the Kawamata-Morrison Cone Conjecture.

Nefness for coherent sheaves. Let us recall that a coherent sheaf E on a normal variety X has a projectivization P(E) with a canonical, so-called tautological, line bundle ζ on it and a natural morphism p : P(E) → X with a natural sheaf quotient map: p * E ↠ ζ. An account on this set-up is given in [START_REF] Dieudonné | Éléments de géométrie algébrique II. Deuxième partie[END_REF]Chapter 4]. We recall the universal property of this construction: for any scheme C with a morphism q : C → X, to give a morphism ν : C → P(E) commuting with the projections to X is equivalent to giving a line bundle L over C together with a sheaf surjection q * E ↠ L.

Projectivizations are standardly used for generalizing positivity notions of line bundles to coherent sheaves, as follows.

Definition 2.21. Let E be a coherent sheaf on a normal variety X. It is called ample, strictly nef, or nef if the tautological line bundle ζ on P(E) is ample, strictly nef, or nef respectively. Remark 2.22. This coincides with [120, Definition 6.1.1] when the sheaf E is locally free. Note that for a torsion-free coherent sheaf E, the scheme P(E) may well have several irreducible components. Somehow, several of these components may be relevant for studying the positivity of E: not only the mere one which is dominant onto X, but also components which may be contracted to a proper non-zero dimensional locus of X. Such components don't exist for a reflexive sheaf on a normal projective surface: so in this case, nefness is easier to study.

Proposition 2.23. We have the following properties:

• if Y ⊂ X is a normal subvariety, and E is a nef coherent sheaf on X, then E| Y is nef [START_REF] Kubota | Ample sheaves[END_REF]Proposition 7];

• conversely, nefness of a coherent sheaf E is enough to be checked on all curves of X [131, Proposition 3.2];

• if f : Y → X is a finite dominant morphism of normal varieties and E is a coherent sheaf on X, E is nef if and only if f * E is; • if f : Y → X is a proper birational morphism resolving the singularities of a normal variety X and E is a coherent sheaf on X such that f * E is nef, then E is nef.
Proof. Let E be a coherent sheaf on a normal variety X and let f : Y → X be a proper dominant morphism. By [49, 4.1.3.1], we have a commutative diagram

P(f * E) π ′ f ′ / / P(E) π Y f / / X with a tautological sheaf compatibility ζ f * E = f ′ * ζ E . Let C be a curve in P(f * E). Then ζ f * E • C = ζ E • f ′ * C, which is positive if E is nef. So if E is nef, f * E is nef.
Assume moreover that f is finite, and let C be a curve in P(E).

Then

ζ E • C = 1 deg f ζ f * E • f ′-1 (C), which is positive if f * E is nef. So if f * E is nef, E is nef.
Lemma 2.24. Let X be a normal projective variety, and Z be a closed proper subscheme of X that is locally generated by a regular sequence. Then the ideal sheaf I Z is not nef.

Proof. Note that the blow-up Bl Z (X) and the projectivization P(I Z ) coincide by [85, 3.10]. Under this identification, the tautological sheaf on P(I Z ) corresponds to the ideal sheaf of the exceptional divisor O Bl Z (X) (-E), by [START_REF]Stacks Project[END_REF]Lemma 31.32.4]. Intersecting with the strict transform of a curve that passes through a point of Z but is not contained in Z, we see that it is not nef.

Remark 2.25. Interestingly enough, the reflexive pullback of a non-nef reflexive sheaf by a finite morphism may be nef, as shows the following example. Let X be the finite quotient of an abelian surface A by the involution i = -id A . Since p : A → X is a finite quasiétale cover, the reflexive sheaves p [ * ] T X and T A are the same, in particular are nef. We are going to prove that T X itself is not nef. By Corollary 2.20, we have

T X = p * (O A ⊕O A ) ⟨-id 2 ⟩ . As the action is diagonal, T X = (p * O A ) ⟨-id 2 ⟩ ⊕(p * O A ) ⟨-id 2 ⟩ . Denote by F the sheaf (p * O A ) ⟨-id 2 ⟩
. We compute locally: let V ⊂ X, U = p -1 (V ) ⊂ A be affine open sets with local coordinates (x, y) ∈ C 2 ≃ U so that p| U ramifies only at (0, 0). The quotient map p : U → V rewrites:

C[u, v, w]/(uv -w 2 ) ∼ = O X (V ) → C[x, y] ∼ = O A (U ) u, v, w → x 2 , y 2 , xy, so its image C[x 2 , y 2 , xy] identifies with the local ring O X (V ). Hence, F(V ) ≃ {f ∈ C[x, y] | ∀ x, y, f (x, y) = -f (-x, -y)} = x C[x 2 , y 2 , xy] ⊕ y C[x 2 , y 2 , xy], so that F ⊗2 (V ) ≃ u O X (V ) ⊕ v O X (V ) ⊕ w O X (V ) = I Sing(X) (V )
. This isomorphism is actually global:

F ⊗2 ∼ = I Sing(X) .
Ideal sheaves are not nef by Lemma 2.24, so F ⊗2 is not nef, so by [119, Proposition 2], F is not nef.

Pseudoeffectivity for reflexive sheaves. For reflexive sheaves, pseudoeffectivity is standardly defined not through projectivization: the main reason for that is that the pushforwards of powers of the tautological sheaf on P(E) are the symmetric powers of E, whereas we are interested in the potential sections of the reflexivized symmetric powers of E.

Definition 2.26. Let E be a reflexive sheaf on a normal projective variety X. It is considered pseudoeffective if there is an ample Cartier divisor H on X such that for all c > 0, there are integers i, j such that i > cj > 0 and

h 0 (X, Sym [i] E ⊗ O X (jH)) ̸ = 0.
Note that if E is pseudoeffective with respect to a certain ample Cartier divisor H, then it is actually pseudoeffective with respect to any ample Cartier divisor [START_REF] Höring | Algebraic integrability of foliations with numerically trivial canonical bundle[END_REF]Lemma 2.3]. Generalizing this definition to any coherent sheaf is not obvious [START_REF] Höring | A non-vanishing conjecture for cotangent bundles[END_REF].

Example 2.27. The sheaf T X in Remark 2.25 is pseudoeffective, as T X = F ⊕ F with S [2] F ∼ = O X . Definition 2.28. Let E be a reflexive sheaf on a normal projective variety X. Denote by P ′ (E) the normalization of the unique dominant component of P(E) onto X. Let P be a resolution of P ′ (E), such that the birational morphism r : P → P ′ (E) over X is an isomorphism precisely over the open locus X 0 ⊂ X reg where E is locally-free.

Denoting by π the morphism P → P(E) and by O P (1) the pullback of the tautological bundle of P(E) by π, [148, V.3.23] asserts that one can choose (often not uniquely) an effective divisor Λ supported in the exceptional locus of r such that

ζ := O P (1) ⊗ O P (Λ) satisfies π * ζ ⊗m ≃ S [m] E for all m ∈ N. Such ζ is called a tautological class of E.
As said in [START_REF] Höring | Algebraic integrability of foliations with numerically trivial canonical bundle[END_REF] Let us fix a c > 0 and take i, j integers such that i > cj > 0 and h 0 (X, S [i] (E) ⊗ O X (jH)) > 0. Up to taking a smaller j (possibly negative), we can assume that

h 0 (X, S [i] (E) ⊗ O X ((j -m)H)) = 0.
By normality of reflexive sheaves,

h 0 (D, S [i] (E| D ) ⊗ O D (jH)) = h 0 (U ∩ D, S i (E| U ∩D ) ⊗ O U ∩D (jH)) ≥ h 0 (U, S i (E| U ) ⊗ O U (jH)) -h 0 (U, S i (E| U ) ⊗ O U ((j -m)H)) = h 0 (X, S [i] (E) ⊗ O X (jH)) -h 0 (X, S [i] (E) ⊗ O X ((j -m)H)) > 0,
where the second equality comes from tensoring by S i (E| U ) ⊗ O U (jH) and going to cohomology in the following exact sequence:

0 → O U (-mH) → O U → O U ∩D → 0.
This result can be slightly generalized. Proposition 2.30. Let X be a normal projective variety, L a globally generated line bundle, E a pseudoeffective reflexive sheaf on X. Then for a very general element D ∈ |L|, the sheaf E| D is reflexive and pseudoeffective.

The idea of this proof stems from Lemma 2.14. By passing through the universal family U for the linear system |L|, it is enough to establish the following two results. Lemma 2.31. Let X and U be normal projective varieties, and let p : U → X be a flat morphism, that is surjective and has connected fibers. If E is a reflexive pseudoeffective sheaf on X, then p * E is a reflexive pseudoeffective sheaf on U .

Proof. By [82, Proposition 1.8], p * E is reflexive, and so is p * S [i] E for any i > 0. In particular, by normality, p * S [i] E = S [i] p * E for any i > 0. Fixing an ample Cartier divisor H on X, and an effective ample Cartier divisor A on U , we set H := p * H + A, an ample Cartier divisor on U . We then fix c > 0, and note that for any i > cj > 0, it holds

h 0 (U, S [i] p * E ⊗ O U (j H)) ≥ h 0 (U, p * (S [i] E ⊗ O U (jH))) = h 0 (X, S [i] E ⊗ O U (jH))
where we use the fact that p is flat, that it satisfies p * O U = O X , and the projection formula. Hence, p * E is pseudoeffective. Lemma 2.32. Let U and Y be normal projective varieties, let q : U → Y be a proper morphism that is surjective and has connected fibers, and whose general fiber is irreducible. If E is a reflexive pseudoeffective sheaf on U , and y is a very general point in Y , then E| q -1 (y) is a reflexive pseudoeffective sheaf on Y .

Proof. By [50, Theorem 12.2.1], for y ∈ Y general, E| q -1 (y) is reflexive, and so is S [i] E | q -1 (y) , which coincides with S [i] E| q -1 (y) by normality. Let us fix an ample divisor H on U . Note that for any positive integer n, there are i > nj > 0 such that S [i] E ⊗ O U (jH) has a non-zero section s n . The set {y ∈ Y | s n | q -1 (y) = 0} is Zariskiclosed in Y , and so for a general y, the reflexive sheaf S [i] E ⊗ O U (jH) | q -1 (y) admits a non-zero section. Hence, for a very general point y ∈ Y , E q -1 (y) is pseudoeffective.

We will use several times the following result [START_REF] Höring | A non-vanishing conjecture for cotangent bundles[END_REF]Lemma 3.15 where M ⊂ N is the set of all m such that mD is Cartier and Bs(mD) is the base locus of the linear system |mD|.

We then define its restricted base locus

B -(D) := n∈N * B Å D + 1 n A ã ,
where A is an arbitrary ample divisor (the locus does not depend on the choice of A by [START_REF] Ein | Asymptotic invariants of base loci[END_REF]Proposition 1.19]).

Of course, a nef Q-divisor has an empty restricted base locus. To that extent, the restricted base locus measures the non-nefness of a pseudoeffective line bundle. However, not all curves of a restricted base locus B -(D) must be D-non-positive, even in the simpler case where D is a pseudoeffective line bundle on a smooth surface and B -(D) is the negative part of its Zariski decomposition.

Stability for a torsion-free sheaf.

Definition 2.35. Let E be a torsion-free coherent sheaf on a normal projective variety X of dimension n. For an ample Q-Cartier divisor H on X, E is said to be Hsemistable, if for some integer m such that mH is Cartier, for all torsion-free non-zero proper subsheaf F ⊂ E, it holds

c 1 (F) • (mH) n-1 rk(F) ≤ c 1 (E) • (mH) n-1 rk(E) .
If the inequality is strict for all such F, we say that E is H-stable.

Let E be a torsion-free coherent sheaf on a normal projective variety X. For any ample Q-Cartier divisor H on X, and for some m big and divisible enough, n -1 general members of |mH| cut out a smooth curve C on which E| C is still torsion free by [50, Theorem 12.2.1], hence locally free. A generalization of a well-known Mehta-Ramanathan result says that stability behaves well under some well-chosen restrictions; we recall it as it is stated in [START_REF] Höring | Algebraic integrability of foliations with numerically trivial canonical bundle[END_REF]Lemma 2.11]. This result shows that, for a locally-free sheaf E, testing stability on the locally-free proper subsheaves is the same as testing stability on the torsion free coherent proper subsheaves: Our notion of (semi)stability is no different from the notion usually defined for locally-free sheaves. Lemma 2.36. Let X be a normal projective variety of dimension n, and H an ample Cartier divisor on X. Let E be a torsion-free coherent sheaf on X, that is stable with respect to H. Then there is m 0 such that, for all m ≥ m 0 , and for D 1 , .

:= p [ * ] E. Then, if E is H-stable, F is p * H-semistable.
Proof. Suppose that E is H-stable. By Lemma 2.36, on a smooth curve C cut out by n -1 very general elements of the linear system defined by a suitable multiple of H, the now locally-free sheaf E| C is still H| C -stable. In particular, [START_REF] Lazarsfeld | Positivity in Algebraic Geometry[END_REF]Lemma 6.4.12] applies; so the pullback sheaf F| p -1 (C) is p * H| C -semistable. Hence, F is H-semistable.

Note that positivity and stability of a zero-slope locally-free sheaf are related by Miyaoka's result [START_REF] Miyaoka | The Chern classes and Kodaira dimension of a minimal variety[END_REF], [START_REF] Lazarsfeld | Positivity in Algebraic Geometry[END_REF]Proposition 6.4.11]: Proposition 2.39. Let E be a vector bundle on a smooth curve C. If E is semistable and c 1 (E) = 0, then E is nef.

More subtle than the mere stability of E is the stability of E and some of its symmetric powers. A first issue is that the torsion freeness of E, an assumption that is crucial to make sense of stability in our definition, does not imply torsion freeness of the symmetric powers of E. Nevertheless, we provide a few results. Lemma 2.40. Let E be a locally free sheaf on a normal projective variety X of dimension n. Suppose that for some positive integer r, the symmetric power S r E is stable. Then E is stable as well.

Proof. Let F be a locally-free non-zero subsheaf of E. Then S r F is a locally free nonzero subsheaf of S r E, and we can compute its slope with respect to an ample Cartier divisor H on X:

c 1 (S r F) • H n-1 rk(S r F) = r c 1 (F) • H n-1 rk(F) .
If F is proper in E, then S r F is proper in S r E; so the stability of S r E implies that of E.

Corollary 2.41. Let E be a reflexive sheaf on a normal projective variety X of dimension n. Suppose that for some positive integer r, the reflexivized symmetric power S [r] E is stable. Then E is stable as well.

Proof. It follows from Lemmas 2.36, 2.40, and from the fact that a reflexive sheaf on a smooth curve is locally-free.

Remark 2.42. We recall an interesting fact stated in [9, Corollary 6, following remark]. If E is a locally-free stable sheaf on a smooth projective variety X, then the following are equivalent

• S r E is stable for some r ≥ 6 ;

• S r E is stable for any r ≥ 6.

Whether or not the stability of all S [r] E for l ∈ N could boil down to the stability of some S [r] E for a finite amount of r's remains an open question.

Nevertheless, this remark allows us to rewrite the result [86, Proposition 1.3] in the following way. Lemma 2.43. Let E be a locally-free sheaf on a smooth curve C. Assume that for some r ≥ 6, S r E is stable, and that c 1 (E) = 0. Denoting by ζ the tautological bundle on P(E), ζ is nef and satisfies:

ζ dim Z • Z > 0,
for any closed proper subvariety Z ⊂ P(E).

We can not emphasize enough that the reflexive pullback p [ * ] E of a H-stable reflexive sheaf E by a finite dominant morphism p is merely p * H-semistable and a priori not stable, let alone his reflexive symmetric powers. However, the conclusive property of Lemma 2.43 is preserved by reflexive pullback. Remark 2.44. Let E be a reflexive sheaf on a normal projective variety X, and C ⊂ X a smooth curve such that E is locally-free in an analytical neighborhood of C, such that the tautological bundle ζ on P(E| C ) is nef and

ζ dim Z • Z > 0
holds for any closed proper subvariety Z ⊂ P(E| C ).

Let p : X → X be a finite dominant morphism, where X is a normal projective variety. Denote Ĉ := p -1 (C), Ê := p [ * ] E and ζ the tautological bundle of P( Ê| Ĉ ). If we have that p * (E| C ) = Ê| Ĉ , then the following diagram is Cartesian with tautological compatibility ζ = q * ζ.

P( Ê| Ĉ ) π q / / P(E| C ) π Ĉ p / / C
Hence, ζ is nef and satisfies, for any closed proper subvariety Z ⊂ P( Ê| Ĉ ):

ζ dim Z • Z > 0.
Remark 2.45. The case in which this remark will be relevant for us is when X is a normal projective surface with an ample Q-Cartier divisor H, C is a smooth curve arising as a very general element of |mH|, for m big and divisible enough, and p : X → X is the morphism constructed in Section 2.6, so that Ê is locally-free. Let X be a normal projective variety, whose singularities in codimension 2 are all quotient singularities. Then X contains a normal quasiprojective subvariety Y with codim X (X \ Y ) ≥ 3 that admits a quasiétale Q-structure, by [START_REF] Greb | The Miyaoka-Yau inequality and uniformisation of canonical models[END_REF]Proposition 3.10]. Let us call the whole data (X, Y, Ŷ , p) an unfolding of X. Note that if (X, Y, Ŷ , p) is an unfolding of a surface X, then Y = X.

V α pα → V α /G α p ′ α → X such that V α is smooth, p ′ α is étale, G α is
Now let E be a reflexive sheaf on X, and let us construct a sheaf Ê that is locally free on an unfolding Ŷ of X. First note that Ŷ is normal, hence Cohen-Macaulay in codimension 2. So, up to removing from Y a finite union of subvarieties of codimension at least 3 in X, we can assume that Ŷ is Cohen-Macaulay. Let We are going to use Ê to define the Chern classes of E. We define a first, a squared first and a second orbifold Chern class of E as multilinear forms on NS(X) n-1 , respectively on NS(X) n-2 by:

ĉ1 (E) • H 1 • • • H n-1 = 1 m n-1 • |G| c 1 ( Ê) • p * (mH 1 ) • • • p * (mH n-1 ), ĉ1 2 (E) • H 1 • • • H n-2 = 1 m n-2 • |G| c 1 ( Ê) 2 • p * (mH 1 ) • • • p * (mH n-2 ), ĉ2 (E) • H 1 • • • H n-2 = 1 m n-2 • |G| c 2 ( Ê) • p * (mH 1 ) • • • p * (mH n-2 ),
where H 1 , . . . , H n-1 are ample Q-classes, and m is big and divisible enough that general elements of p * (mH 1 ), . . . , p * (mH n-1 ) cut out a complete intersection smooth curve in Ŷ and general elements of p * (mH 1 ), . . . , p * (mH n-2 ) a complete intersection normal surface in Ŷ .

As stated in [75, Theorem 3.13.2], these orbifold Chern classes are compatible with general restrictions, and so is the unfolding construction [START_REF] Greb | The Miyaoka-Yau inequality and uniformisation of canonical models[END_REF]Proposition 3.11].

If X is a normal klt variety, we denote by ĉ1 (X), ĉ1 2 (X), ĉ2 (X) the corresponding Chern classes for the reflexive sheaf T X .

Example 2.48. Let X be a smooth projective variety of dimension n, G be a finite group acting on X freely in codimension 1. Then (X/G, X/G, X, p) is an unfolding of X/G, and

T X = p [ * ] T X/G , so ĉ2 (X/G) • H 1 • • • H n-2 = c 2 (X) • p * H 1 • • • p * H n-2 .
for any ample divisors H 1 , •H n-2 on X/G. In particular, if Z is a finite quasiétale quotient of a torus, then ĉ2 (Z)

• H 1 • • • H n-2 =
(3c 2 (X) -c 1 (X) 2 ) • H 1 • . . . • H n-2 ≥ 0.
A partial equality case in Theorem 2.51 is entirely characterized.

Theorem 2.52. [210, Theorem 1.1], [77, p.4-5] Let X be a smooth projective variety of dimension n. Suppose that c 1 (X) = 0 and that for some ample divisor H, c 2 (X)

• H n-2 = 0. Then X is uniformized by C n .
Let us discuss how this picture generalizes to the singular setting. Several results build up to generalize Theorem 2.51. The first one is in a very singular setting, in dimension 2. It was used, as we advertised, to answer a boundedness question in [START_REF] Moraga | Bounding singular surfaces via Chern numbers[END_REF].

Theorem 2.53. [138, Theorem 0.1] Let X be a normal projective surface with log canonical singularities such that κ(X) ≥ 0. Then 3 ĉ2 (X) -ĉ1 2 (X) ≥ 0.

The second one is in a mildly singular setting, but in dimension three.

Theorem 2.54. [173, Theorem 1.2] Let X be a normal projective threefold with isolated log canonical singularities such that K X is movable. Then for any ample divisor H on X, 3c 2 (X)

• H -c 1 2 (X) • H ≥ 0.
The third one is in arbitrary dimension, and in the klt setting.

Theorem 2.55. [START_REF] Guénancia | Orbifold stability and Miyaoka-Yau inequality for minimal pairs[END_REF]Theorem B] Let X be a normal projective variety of dimension n with klt singularities such that K X is nef. Let m be a positive integer such that mK X is Cartier. Let ν be the largest integer such that (mK X ) ν is not numerically trivial. Then, for i = min(ν, n -2) and for

j = n -2 -i, it holds Å ĉ2 (X) - n 2(n + 1) ĉ1 2 (X) ã • (mK X ) i • H j ≥ 0.
Note that those generalizations are initially stated in the broader context of pairs with standard coefficients. As in Theorem 2.52, the partial equality case is classified. Theorem 2.56. [START_REF] Lu | A characterization of finite quotients of abelian varieties[END_REF]Theorem 1.2] Let X be a normal projective variety of dimension n. Then X is a quotient of an abelian variety by a finite group action that is free in codimension 1 if and only if X is klt, c 1 (X) = 0, and for some ample divisors

H 1 , . . . , H n-2 on X, ĉ2 (X) • H 1 • • • H n-2 = 0.
Let us close this section with two inequalities that have little to do with minimal varieties. As we discussed in Example 2.48, the second orbifold number of the singular Kummer surface vanishes, whereas its resolution, a K3 surface, has Euler number 24. This phenomenon generalizes into the following inequality, which will be important to prove Theorem 2.119, a motivation for Part II. A similar inequality is established in [START_REF] Greb | Harmonic metrics on Higgs sheaves and uniformization of varieties of general type[END_REF]Claim 7.1].

Theorem 2.57. Let X be a normal klt variety of dimension n, and let ε : X → X be a resolution of X. For any ample divisor H on X, it holds

c 2 ( X) • (ε * H) n-2 ≥ ĉ2 (X) • H n-2
and equality occurs if and only if X is smooth in codimension 2.

Proof. This is claimed in dimension 3 by [182, Proposition 1.1], and [START_REF] Lu | A characterization of finite quotients of abelian varieties[END_REF]Remark 1.5] claims that it generalizes it to arbitrary dimension, but let us include a proof, as we could not entirely follow [START_REF] Shepherd-Barron | Singular threefolds with numerically trivial first and second Chern classes[END_REF].

First, up to replacing H be a large multiple of itself, we can assume that it is very ample. Let S be a normal complete intersection surface cut out by n -2 general members of the linear system |H|. Let S ′ be its strict transform in X, which is in fact cut out by the strict transforms of the n -2 general members of |H|, i.e., by n -2 general members of ε * |H|. We are left showing that

c 2 (T X | S ′ ) ≥ ĉ2 (T X | S ).
Consider the local diagram of an unfolding of X.

Vα i α pα / / V α p α / / V α /G α p ′ α Ŷ p / / Y
Note that by [START_REF] Greb | The Miyaoka-Yau inequality and uniformisation of canonical models[END_REF]Proposition 3.11], there is a compatible unfolding of S, namely Ŵα

i α pα / / W α p α / / W α /G α p ′ α Ŝ p / / S
where W α ⊂ V α is cut out by n -2 general members in the linear system p α * p ′ α * |H|. Note that V α and W α are smooth and

p α [ * ] p ′ α * T S | pα(Wα)∩S = T Wα p α [ * ] p ′ α * T X | pα(Vα)∩S = T Vα | Wα
So we get an exact sequence of locally free sheaves on

W α 0 → p α [ * ] p ′ α * T S | pα(Wα) → p α [ * ] p ′ α * T X | pα(Vα)∩S → p α * p ′ α * O S (H) ⊕n-2 → 0
As pullback is exact for locally free sheaves, we can pullback by pα and glue the locally free sheaves obtained onto Ŝ, to get

0 → p [ * ] T S → p [ * ] T X | S → p * O S (-mH) ⊕n+2 → 0, hence ĉ2 (T X | S ) = ĉ2 (S) + Ç n -2 2 å m 2 (H| S ) 2 + (n -2)m ĉ1 (S) • H| S .
Since X and S ′ are smooth, we also have

0 → T S ′ → T X | S ′ → O S ′ (ε * (mH)) ⊕n-2 → 0 so c 2 (T X | S ′ ) = c 2 (S ′ ) + Ç n -2 2 å m 2 (ε * H| S ′ ) 2 + (n -2)mc 1 (S ′ ) • ε * H| S ′ .
We are left proving that c 2 (S ′ ) = e(S ′ ) ≥ e orb (S) = ĉ2 (S), but this follows from the exceptional locus in S ′ being a union of trees of P Finally, this last inequality states that the Chern numbers of a positive vector bundle are, in some sense, positive. It will come handy in Sections 4.1 and 24.

Theorem 2.58. [START_REF] Demailly | Compact complex manifolds with numerically effective tangent bundles[END_REF]Corollary 2.6] Let X be a smooth projective variety of dimension n, and E be a nef vector bundle on X. Then for any 1 ≤ r ≤ n, any Chern monomial

c i 1 (E) • • • c i k (E) with i 1 + . . . + i k = r and any ample line bundle H on X satisfy c 1 (E) r • H n-r ≥ c i 1 (E) • • • c i k (E) • H n-r ≥ 0.

An introduction to valuation theory for singularities. Recall that an integral valuation

v on a field K is a function ν : K → Z ∪ {+∞} that satisfies, for all a, b ∈ K, • v(a) = +∞ if and only if a = 0; • v(a + b) ≥ min(v(a), v(b)); • v(ab) = v(a) + v(b).
A discrete valuation is an integral valuation which is surjective onto Z ∪ {+∞}.

Example 2.59. We say that E is a divisor over a normal complex analytic variety X if there is a partial resolution of X, i.e., a normal complex analytic variety X with a proper birational morphism φ : X → X, such that E is a φ-exceptional prime divisor. We say that the partial resolution φ realizes E. To such a divisor we associate a discrete valuation on the function field of X:

v E : f ∈ k(X) → ord E (f • φ) ∈ Z ∪ {+∞},
which does not depend on the partial resolution φ chosen. A valuation of this form is called a divisorial valuation. To propose a second example, we first need to introduce the notion of a junior element. It will play an important role in Part II. Definition 2.60. Let g be a matrix in GL n (C). Assume that it has finite order d. Since g d = id, g is diagonalizable and has eigenvalues of the form e 2iπa k /d , for integers

a k ∈ [[0, d -1]].
The age of g is set to be the number a 1 +...+an d . If it equals 1, we say that g is junior.

Example 2.61. [START_REF] Ito | The McKay correspondence for finite subgroups of SL(3, C). In Higher-dimensional complex varieties[END_REF] Let g ∈ SL n (C) be a matrix of finite order d. We can take coordinates x 1 , . . . , x n on C n that diagonalize g, so that for any k

∈ [[1, n]], g * x k = e 2iπa k /d x k , with a k ∈ [[0, d -1]].
We define the integral valuation:

v g : x k ∈ k(C n ) → a k ∈ Z ∪ {+∞}.
If g is junior, then a 1 , . . . , a n have no common prime divisor, and thus v g is then a discrete valuation.

The following theorem has a reinterpretation in terms of valuations.

Theorem 2.62. [START_REF] Ito | The McKay correspondence for finite subgroups of SL(3, C). In Higher-dimensional complex varieties[END_REF] Let C n /G be a finite Gorenstein quotient singularity, and let X be a terminalization of it. Then there is a natural one-to-one correspondence between conjugacy classes of junior elements in G and prime exceptional divisors in X.

Remark 2.63. Note that the correspondence in Theorem 2.62 is just the identification of the set of divisorial valuations v E , when E is a crepant divisor over C n /G, and the set of valuations v g , when g is a junior element in G. Definition 2.64. Let X, Y be normal complex analytic varieties, and p : X → Y be a finite Galois morphism of group G. Let v, w be discrete valuations on the function fields k(X) and k(Y ). Note that k(Y ) identifies with the invariant subfield k(X) G of k(X). The ramification index Ram(v, k(Y )) of v over k(Y ) is the unique non-negative integer such that:

v(k(Y ) * ) = Ram(v, k(Y ))Z.
We say that v is an extension of w to k(X) if:

w = 1 Ram(v, k(Y )) v | k(Y ) . If v is
E. Hence v F | H Y,E = rv E , i.e., v F is an extension of v E to k(X) with ramification index Ram(v F , k(Y )) = r.
Conversely, if we assume that v F is an extension of v E to k(X), then the structure sheaf map p ♯ : H Y → p * H X sends the ideal sheaf I E to a subsheaf of p * I F , so F dominates E.

Another important concept when considering ramification of valuations over subfields is the following. Definition 2.66. Let X, Y be normal complex analytic varieties, and p : X → Y be a finite Galois morphism of group G. Let v be a discrete valuation on k(X). Let R v ⊂ k(X) be the valuation ring, and m v ⊂ R v be the unique maximal ideal. We define the inertia group

G T (v) := {g ∈ G | ∀ x ∈ R v , gx -x ∈ m v }.
G T (v h ) = G ∩ ⟨h⟩.
2.9 Automorphisms of abelian varieties. Let us first recall the basic definitions.

Definition 2.69. A complex torus is a quotient of the form C n /Λ, where Λ is a lattice in C n . A complex torus of dimension n = 1 is called an elliptic curve. Definition 2.70. An abelian variety A is a complex torus that admits a holomorphic embedding into a projective space. Definition 2.71. We denote by Aut(A) the set of biholomorphisms from A to A. We call them automorphisms of A. The starting point when studying automorphisms of abelian varieties is the following proposition. Proposition 2.74. [17, Proposition 1.2.1] Let X = C n /Λ be a complex torus. Let g : X → X be a holomorphic map. Then there exists a unique matrix M (g) ∈ GL n (C) and a unique point T (g) ∈ X such that M (g)(Λ) ⊂ Λ and, for all x ∈ X, g(x) = M (g)x + T (g).

Whereas every point defines a translation, not every matrix defines a holomorphic map on a torus. There is namely the following constraint. Proposition 2.75. [17, Proposition 1.2.3] Let X = C n /Λ be a complex torus. Let g : X → X be a holomorphic map. Then the characteristic polynomial of M (g)⊕M (g) has rational coefficients.

We derive the following corollary, that will be used extensively in Part II. Lemma 2.76. Let A be an abelian variety of dimension n, and g ∈ Aut(A) of finite order. Denote by P (g) the characteristic polynomial of M (g). Then P (g)P (g) is a product of cyclotomic polynomials.

Proof. By [17, Proposition 1.2.3], P (g)P (g) is a polynomial over Q. Since g has finite order, the roots of this polynomial are roots of unity. Remembering that cyclotomic polynomials are the minimal polynomials of roots of unity over Q, an easy induction shows that there is a product Π of cyclotomic polynomial that has the exact same roots as P (g)P (g). But since both cyclotomic polynomials and characteristic polynomials are unitary, it means that P (g)P (g) = Π.

Even with this constraint, not every matrix can act on every abelian variety.

Example 2.77. [START_REF] Birkenhake | Complex abelian varieties[END_REF]Corollary 13.3.4] Let E be an elliptic curve. If M (g) is an automorphism of E of finite order d ≥ 3 such that M (g)(0) = 0, then we are in one of the folowing three cases:

d 3 4 6 M (g) j i -j E E j E i E j
where j = e 2iπ/3 and E z = C/(Z + zZ).

We will discuss partial results in higher dimension in Part II, in the following direction: knowing the matrix of an automorphism of A, what can we say about A? This has to do with the theory of abelian varieties with complex multiplication (CM). We will recall some background on them and a useful proposition, following [17, 13.3] and [START_REF] Shimura | Complex multiplication of abelian varieties and its applications to number theory[END_REF].

Definition 2.78.

A number field is totally real if for every embedding of it into the complex numbers, its image lies in the real line. It is totally complex if it cannot be embedded into the real numbers.

Definition 2.79.

A CM-field is a totally complex quadratic extension of a totally real number field.

Example 2.80. Clearly, Q[j] and Q[i] are CM-fields. Every cyclotomic field Q[ζ n ] is clearly totally complex, and is in fact a CM-field. Indeed, the number field Q[ζ n + ζ -1
n ] is totally real, and ζ n is a root of the quadratic equation

(2X -ζ n -ζ n -1 ) 2 -(ζ n + ζ n -1 ) 2 + 4 = 0.
More importantly for us, defining the following quadratic integers

u 7 = -1 + i √ 7 2 , u 8 = i √ 2, u 15 = 1 + i √ 15 2 , u 20 = i √ 5, u 24 = i √ 6,
and the following algebraic integer, whose square is a quadratic integer

u 16 = i » 4 + 2 √ 2.
then clearly, for each k ∈ {7, 8, 15, 16, 20, 24}, Q[u k ] is a CM-field.

Definition 2.81.

A CM-type of a CM-field K of degree 2g over Q is the data of embeddings {σ 1 , . . . , σ g } of K into C that are pairwise distinct and pairwise nonconjugated.

Example 2.82.

There are exactly two CM-types for Q[j]: either the natural embedding, or the conjugated embedding. An example of a CM-type for Q[ζ 7 ] is given by

σ 1 :ζ 7 → ζ 7 σ 2 :ζ 7 → ζ 7 2 σ 3 :ζ 7 → ζ 7 4
It has in fact eight different CM-types.

Definition 2.83. Given a CM-field K and a CM-type Σ = {σ 1 , . . . , σ g }, an abelian variety of CM-type (K, Σ) is an abelian variety A of dimension g such that there is an embedding ρ : K → End Q (A) such that, when taking matrices,

M • ρ : K → M g (C) is conjugated in GL g (C) to σ 1 ⊕ . . . ⊕ σ g ,
Q • O := ® n i=1 q i o i | q i ∈ Q, o i ∈ O, n ∈ N ´= A.
Example 2.87. By [START_REF] Reiner | Maximal orders[END_REF]Chapter 12], if K is a number field, viewing it as a Q-algebra, its ring of integers is the unique maximal order in it.

Definition 2.88. An abelian variety

A of CM-type (K, Σ) is called principal if End(A)∩ ρ(K) is the maximal order in ρ(K) ≃ K. Remark 2.89. If A is an abelian variety of CM-type (K, Σ) such that the embedding K ⊂ End Q (A) sends the ring of integers of K to a subring of End(A), then A is principal. Example 2.90. The elliptic curve E j is of CM-type (Q[j], σ). It is principal because End(E j ) = Z[j] is a maximal order in Q[j]. The elliptic curve E 2j is of CM-type (Q[j], σ) too. It is not principal because End(E 2j ) = Z[2j] is not a maximal order in Q[j].
Proposition 2.91. [START_REF] Shimura | Complex multiplication of abelian varieties and its applications to number theory[END_REF]p.60,Proposition 17] The number of principal non-isomorphic abelian varieties of a same CM-type (K, Σ) is the class number of K. In particular, there is exactly one if and only if the ring of integers of K is a principal ideal domain. Definition 2.92. An abelian variety is said to be simple if it contains no proper abelian subvariety of positive dimension. A CM-type is called primitive if every abelian variety of this CM-type (or equivalently, one abelian variety of this CM-type) is simple.

We can determine purely algebraically whether a given CM-type is primitive. Definition 2.93. Let K 0 be a totally real number field. An element x ∈ K 0 is totally positive if for every embedding K → R, the image of x is a positive number.

Example 2.94. The unit 1 is totally positive, -1 is clearly not totally positive, and Clearly, all ratios are negative, except for v 16 u 16 = √ 2 -1. But as we can define an embedding K 0 → R that sends √ 2 to -√ 2, and as -√ 2 -1 now is negative, the ratio v 16 u 16 is not totally positive. So setting Σ as the set of embeddings σ : K → C such that Im(σ(u 16 )) > 0, we obtain a primitive CM-type (K, Σ). As K has degree 4 over Q, the corresponding simple abelian varieties have dimension 2.

1 + √ 2 ∈ Q[ √ 2] is positive in the natural embedding, but not totally positive as 1 - √ 2 < 0 is its image by the conjugated embedding of Q[ √ 2] in R.
• -ζ 2 = η; • K 0 [ζ] = Q[ζ] = K; • Σ is the set of embeddings σ : K → C such that Im(σ(ζ)) > 0; • for all ζ ′ ̸ = ζ conjugated to ζ over Q, the ratio ζ ′ ζ -1 is not totally positive.
Finally, we recall without proof a result [184, p.46, Proof of Theorem 3] that we use in Part II. Lemma 2.97. Let K = Q(α) be a totally imaginary quadratic extension of Q of degree 2m. Let F be a finite Galois extension of K, of degree 2r over Q. Let {φ i } 1≤i≤r be morphisms of Q-algebras defined from F to C such that:

Hom Q-alg (F, C) = Vect Q (φ 1 , φ 1 , . . . , φ r , φ r ) .

Suppose also that no two of the restrictions φ

i | K are conjugate.
Then we can restrict m of these morphisms, defining ψ j = φ i j | K for some distinct i j with j ∈ [ [1, m]], such that:

Hom Q-alg (K, C) = Vect Q ψ 1 , ψ 1 , . . . , ψ m , ψ m . We obtain a Z-algebra ∆ := Z[(ψ 1 (α), . . . , ψ m (α))] that is a lattice of rank 2m in C m . The complex torus A := (C m /∆) n/m is an abelian variety of CM-type (F, {φ i } 1≤i≤f ).

A reminder in Sylow theory.

As we will use them thoroughly in Part II, we recall a few facts about p-groups and the three Sylow theorems. Definition 2.98. [START_REF] Rotman | An introduction to the theory of groups[END_REF]Corollary 4.3] Let p be a prime number. A p-group is a group in which every element has order a power of p. Equivalently, it is a group whose order is a power of p.

We use the following result to bound the p-Sylow subgroups of a group of matrices acting on an abelian variety, in which commutation is rare: see the proofs in Chapter 10 for more precise statements.

Theorem 2.99. [START_REF] Rotman | An introduction to the theory of groups[END_REF]Theorem 4.4] Let p be a prime number. Let S be a finite nontrivial p-group. Then the center Z(S) of S contains an element of order p.

A corollary is the following result.

Corollary 2.100. Let p be a prime number. Let S be a finite p-group. If the maximal abelian subgroup of S is cyclic of order p, then S itself is cyclic of order p.

Note that, although we do not need this stronger result in this thesis, the following nice generalization holds, bounding the order of a p-group in which commutation is rare. In particular, all p-Sylow subgroups have the same order, determined by the first Sylow theorem. Theorem 2.106. [START_REF] Rotman | An introduction to the theory of groups[END_REF]Theorem 4.12] Let p be a prime number. Let G be a finite group. Then the number n p of p-Sylow subgroups of G is congruent to one modulo p and divides |G|.

A last important point is the following theorem about finding a normal complement to a Sylow subgroup. Definition 2.107. Let G be a finite group, and S be a p-Sylow subgroup in G. We say that S admits a normal complement if there exists a normal subgroup

N ◁ G such that G ∼ = N ⋊ S.
It is important to note that if G has to be generated by elements of order prime to p, then a p-Sylow subgroup of G cannot have a normal complement. Indeed, if G ∼ = N ⋊ S, all elements of order prime to p belong to N , therefore generate a subgroup of N , which is proper in G.

The following result yields the existence of a normal complement in some cases.

Theorem 2.108. [START_REF] Rotman | An introduction to the theory of groups[END_REF]Theorem 7.50] Let G be a finite group and let p be a prime number dividing the order of G. Let S be a p-Sylow subgroup of G. If S is abelian and

N G (S) = C G (S)
, then S admits a normal complement.

A primer on Mori theory.

Mori theory is a vast branch of birational geometry. For our purpose though, only results tracing back to the eighties will be needed, namely the base point free theorem, a few theorems around contractions, and the Cone Theorem. We follow [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF] in the treatment of these results, but the reader might alternatively refer to [START_REF] Debarre | Higher-dimensional algebraic geometry[END_REF]. Let us first present the base point free theorem, and the contraction theorems.

Theorem 2.109. Base point free Theorem [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]Theorem 3.3] Let (X, B) be a projective klt pair, and let D be a nef Cartier divisor on X such that for some positive rational number a, the divisor aD -(K X + B) is nef and big. Then there exists m 0 such that, for all m ≥ m 0 , the linear system |mD| is base point free.

Definition 2.110. A contraction is a proper morphism of normal varieties

h : X → Y such that h * O X = O Y .
Equivalently, it is a proper surjective morphism with connected fibers.

Definition 2.111. The Mori cone N E(X) of a normal projective variety X is the closure of the cone generated by classes of effective 1-cycles in N 1 (X).

Example 2.112. [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]Example 1.23(4)] The Mori cone of P 2 blown-up in the 9 base points of a general pencil of cubics, whose members are all ireducible, is a tendimensional cone generated by infinitely many classes of (-1)-curves, which correspond to sections of the anticanonical elliptic fibration onto P 1 .

In general, describing the Mori cone of a given variety is strenuous, and equivalent to describing its dual cone, i.e., the nef cone defined in Section 2.4. However, describing the contractions of a variety X allows to say some things about part of its Mori cone, as the Cone Theorem will soon state formally. Before that, let us describe some special contractions in more detail.

Definition 2.113. An extremal ray R of a closed convex cone C is a halfline R ⊂ C such that for all u, v ∈ C with u + v ∈ R, u, v belong to R. It is called K X -negative if there exists Z ∈ R such that K X • Z < 0.
Definition 2.114. [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]Definition 1.25] Let X be a normal projective variety, and R be an extremal ray of

N E(X). A contraction associated to R is a contraction h : X → Y such that for every curve C in X, h(C) = {pt} if and only if [C] ∈ R. Definition 2.115. A contraction of a K X -negative extremal ray is called a Mori contraction.
Example 2.116. Consider the surface S that is P 2 blown-up in two distinct, non infinitesimally close points, and let ℓ ⊂ S be the strict transform of a line through these two points. Then ℓ is a K S -negative (-1)-curve on S, hence it spans a K Snegative extremal ray [42, Lemma 6.2(b)]. There is a Mori contraction associated to ℓ, which is a map S → P 1 × P 1 . Theorem 2.117. Contraction Theorem [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]Theorem 3.7(3) and ( 4)], [START_REF] Debarre | Higher-dimensional algebraic geometry[END_REF]Theorem 7.39] Let (X, B) be a normal projective variety with klt singularities, and let R be a K X + B-negative extremal ray of N E(X). Then there is a contraction h : X → Y associated to R, and moreover

• -(K X + B) is h-ample; • if D is a Cartier divisor and for all curve C contracted by h, it holds D • C = 0, then there exists E a Cartier divisor on Y such that D ∼ h * E; • ρ(X) = ρ(Y ) + 1.
A little more can be said about X and Y when h : X → Y is a contraction that is divisorial, i.e., it is a birational contraction and its exceptional locus has a component of codimension 1 in X. A birational contraction that is not divisorial is called a small contraction.

Proposition 2.118. [111, Proposition 2.5] Let (X, B) be a normal projective Qfactorial pair, and let h : X → Y be a divisorial contraction of a (K X + B)-negative rational extremal ray. Then the exceptional locus of h consists in one irreducible divisor E.

An important corollary of these theorems is the following fact presented in the introduction of Part II. Let us present a detailed proof of it.

Theorem 2.119. Let X be a Calabi-Yau manifold of dimension n. The following are equivalent:

(i) There is a nef and big divisor D on X such that c 2 (X) • D n-2 = 0.

(ii) There is an abelian variety A and a finite group G acting freely in codimension

2 on A such that X is a crepant resolution of A/G.

If it satisfies these conditions, X is called a Calabi-Yau manifold of type n 0 .

Proof. First assume that there is a nef and big divisor D on X such that c 2 (X)•D n-2 = 0. Then by the base point free theorem, as K X is trivial, there is an integer m such that |mD| is base point free. This linear system induces a morphism ϕ : X → P(H 0 (X, mD) * ), whose image we denote by Y . Let H be an ample divisor on X, let H ′ be an ample Cartier divisor on Y such that ϕ * H ′ = D. As D is big, the variety Y has dimension n. Thus, ϕ is a birational contraction. Let us follow [START_REF] Uehara | Calabi-Yau threefolds with infinitely many divisorial contractions[END_REF] and say more about ϕ. Note that for j large enough, jD -H is big (as the big cone is open). Hence for k large enough, both jkD -kH and (k -1)H are effective. So jkD -H is effective. Moreover, it is ϕ-antiample. So, for ε > 0 small enough, setting B := ε(jkD -H), the pair (X, B) is klt by [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]Corollary 2.35(2)] and ϕ is a (K X + B)-negative contraction.

Note that as

K X is trivial, ϕ * ω X = ϕ * O X = O Y . Moreover, (ϕ * ω X ) * *
and ω Y are reflexive sheaves which coincide in codimension 1, so by normality of reflexive sheaves, they coincide. Hence ω Y ≃ O Y . In particular, Y has canonical singularities. Moreover, by Theorem 2.55 and Theorem 2.57,

0 = c 2 (X) • D n-2 ≥ ĉ2 (Y ) • H ′ n-2 ≥ 0.
As equality holds, Y is smooth in codimension 2. Moreover, by Theorem 2.56, Y is a finite quotient of an abelian variety A by a group G acting freely in codimension 1. As G contains no pseudoreflection, A/G sing is exactly the union of the fixed loci of the element of G, so smoothness in codimension 2 implies that G acts freely in codimension 2.

Conversely, assume that X is a crepant resolution of A/G, with G acting freely in codimension 2. Denote by H ′ an ample divisor on A/G, by D its pullback to X. Clearly, D is nef and big. By the equality case in Theorem 2.57, c 2 (X)

• D n-2 = c 2 (A/G) • H ′ n-2 = 0. This concludes the proof.
We can now conclude this exposition with the Cone Theorem.

Theorem 2.120. [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]Theorem 3.7(1) and ( 2)] Let (X, B) be a normal projective klt pair. Then there are countably many rational curves

C i ⊂ X such that 0 < -(K X + B) • C i ≤ 2 dim(X) and N E(X) = N E(X) K X +B≥0 + i∈I R + [C i ].
Moreover, for any ample divisor H and for any ε > 0, there exists

I 0 ⊂ I finite such that N E(X) = N E(X) K X +B+εH≥0 + i∈I 0 R + [C i ].
2.12 Families of deformations of rational curves. In this section, we work with the scheme RatCurves n (X), which is the normalization of the scheme parametrizing rational curves in X. As a rational curve is a 1-cycle, we also work with the scheme Chow(X) parametrizing 1-cycles in X, and the natural map RatCurves n (X) → Chow(X).

ϕ f ′ / / Univ chow (X) π / / X RatCurves n (X) f / / Chow(X)
where ϕ is a P 1 -bundle, and for every rational curve C in X, there exists (finitely many and at least) a point p ∈ RatCurves n (X) such that ev(ϕ

-1 (p)) = C.
If C is a rational curve in X, we may denote by V a family of deformations of C, that is an irreducible component of RatCurves n (X) containing a point corresponding to C. We then define Locus(V) := ev(ϕ -1 (V)) ⊂ X.

We say that the family V is covering if Locus(V) = X. We say that the family V is

unsplit if V is proper over Spec(C), or equivalently if f (V) is closed in Chow(X).
For x ∈ Locus(V), we define V x := ϕ(ev -1 (x)) the family of deformations of C through x. We finally define Locus(V x ) := ev(ϕ -1 (V x )) ⊂ X.

We will use the following lemma repeatedly.

Lemma 2.122. [108, Corollary IV.2.6] Let X be a smooth projective variety. Let V be a family of deformations of a rational curve C in X. If V is unplit, then dim Locus(V) + dim Locus(V x ) ≥ -K X • C + dim(X) -1.
The following lemma produces unsplit families of rational curves. Lemma 2.123. Let X be a smooth projective variety. Suppose that -K X • C > 0 for every rational curve C ⊂ X. Suppose that X has a fibred Mori contraction π : X → Y with dim Y > 0, and let C be a rational curve such that π(C) ̸ = {pt} and such that

-K X • C = min{-K X • B | B rational curve in X, π(B) ̸ = {pt}}.
Then the family of deformations of C is unsplit.

Proof of Lemma 2.123. Let V be the family of deformations of C. Suppose that it is splitting, i.e., C ≡ num i

a i C i ,
with rational curves C i and coefficients

a i ≥ 1 such that i a i ≥ 2. Since -K X is positive on rational curves, we have -K X • C i < -K X • C for all i. So, by minimality of -K X • C, the fibration π contracts all curves C i . Let H be an ample divisor on Y . We obtain i a i C i • π * H = 0, contradiction.
The interaction of fibred Mori contractions and unsplit families of rational curves is especially nice, as the following lemma says. Lemma 2.124. Let X be a smooth projective variety. Suppose that X has a fibred Mori contraction π : X → Y with dim Y > 0, and let C be a rational curve such that π(C) ̸ = {pt} and such that its family of deformations V is unsplit. Then, for any

x ∈ Locus(V), dim Locus(V x ) ≤ dim Y.
Proof of Lemma 2.124. We claim that π| Locus(Vx) is finite onto its image. If it is not, it contracts a curve B to a point: for some ample divisor H on Y , we have

B • π * H = 0. By [3, Lemma 4.1], the numerical class of B ⊂ Locus(V x ) is a multiple of C ∈ N 1 (X) Q , whence C • π * H = 0, which is a contradiction. So π| Locus(Vx) is finite onto its image: this implies dim Locus(V x ) ≤ dim Y .

INTRODUCTION

Complex algebraic varieties with trivial canonical class are of great importance in birational geometry. Indeed, they appear naturally as possible minimal models in the Minimal Model Program (MMP), and come in quite diverse geometrical families. Since higher-dimensional MMP generally gives rise to singular minimal models, understanding singular projective varieties with trivial canonical class is particularly relevant. Recently, three papers [86, Theorem 1.5], [START_REF] Greb | Klt varieties with trivial canonical class. Holonomy, differential forms and fundamental groups[END_REF], [START_REF] Druel | A decomposition theorem for singular spaces with trivial canonical class of dimension at most five[END_REF] achieved a singular decomposition result for these varieties :

Theorem 3.1. Let X be a normal projective variety with klt singularities, with K X numerically trivial. Then there exists a normal projective variety X with at most canonical singularities, which comes with a quasiétale finite cover f : X → X and decomposes as a product:

X ∼ = A × i Y i × j Z j ,
where A is a smooth abelian variety, the Y i are singular Calabi-Yau varieties and the Z j are singular irreducible holomorphic symplectic (IHS) varieties, as defined in Section 5.2.

May it seem an expected generalization of the smooth Beauville-Bogomolov decomposition result [START_REF] Beauville | Variétés kählériennes dont la première classe de Chern est nulle[END_REF], [START_REF] Beauville | Some remarks on Kähler manifolds with c 1 = 0[END_REF], this theorem however relies on serious results from each paper: [START_REF] Greb | Klt varieties with trivial canonical class. Holonomy, differential forms and fundamental groups[END_REF] introduces algebraic holonomy and studies infinitesimal decompositions of the tangent sheaf T X ; [START_REF] Druel | A decomposition theorem for singular spaces with trivial canonical class of dimension at most five[END_REF] deals with the abelian part in the infinitesimal decomposition through a positive characteristics argument, and proves an integrability criterion for the remaining subsheaves of T X ; [START_REF] Höring | Algebraic integrability of foliations with numerically trivial canonical bundle[END_REF] establishes positivity results which add up to Druel's criterion to finish the proof. This proof was notably simplified by [START_REF] Campana | The Bogomolov-Beauville-Yau decomposition for klt projective varieties with trivial first Chern class -without tears[END_REF], shortcutting the positive characterictics argument. Furthermore, the two recent papers [START_REF] Claudon | Kähler spaces with zero first Chern class: Bochner principle, fundamental groups, and the Kodaira problem[END_REF], [START_REF] Bakker | Algebraic approximation and the decomposition theorem for Kähler Calabi-Yau varieties[END_REF] extend this decomposition result to the singular Kähler case by subtle algebraic approximation considerations.

Interestingly enough, the singular decomposition for a klt variety X may not be the same as the singular decomposition of its terminalisation. The typical example is a singular Kummer surface, which resolves by 16 blow-ups into a smooth K3 surface, but has the Beauville-Bogomolov type of an abelian surface. Other such intriguing examples are given in [START_REF] Greb | Klt varieties with trivial canonical class. Holonomy, differential forms and fundamental groups[END_REF]Sect.14]. Compatibility of the singular Beauville-Bogomolov decomposition with terminalisation nevertheless holds for some klt varieties with trivial canonical class [55, Lemma 4.6]. This license to terminalise is essential in the current proof of [86, Theorem 1.5], as it involves positivity results [START_REF] Höring | Algebraic integrability of foliations with numerically trivial canonical bundle[END_REF]Theorem 1.1] for klt varieties which are smooth in codimension 2: any klt variety is not, but its terminalisation surely is.

Since these positivity results have a wider scope than the mere proof of the singular decomposition theorem, it is worth extending them to normal projective klt varieties. Our main theorem is: Theorem 3.2. Let X be a normal projective variety with klt singularities and numerically trivial K X . If its tangent or reflexivized cotangent sheaf is pseudoeffective, then there is a quasiétale finite cover X → X such that q( X) ̸ = 0. Equivalently, the singular Beauville-Bogomolov decomposition of X has an abelian factor of positive dimension.

In particular, if X is a singular Calabi-Yau or IHS variety in the sense of Def.5.2, then neither T X nor its dual Ω [1] X is pseudoeffective.

Importantly enough, this theorem does not boil down to [START_REF] Höring | Algebraic integrability of foliations with numerically trivial canonical bundle[END_REF]Theorem 1.6] on a terminalisation of X; we inevitably have to deal with codimension 2 quotient singularities on X. In this perspective, we resort to the theory of orbifold Chern classes. It has been developped in the late eighties in connection to the abundance problem for threefolds [START_REF] Kollár | Flips and abundance for algebraic threefolds -A summer seminar at the university of Utah[END_REF], and we will extensively use some of its most recent developments, inter alia [START_REF] Lu | A characterization of finite quotients of abelian varieties[END_REF], [START_REF] Greb | The Miyaoka-Yau inequality and uniformisation of canonical models[END_REF], [START_REF] Greb | Projective flatness over klt spaces and uniformisation of varieties with nef anticanonical bundle[END_REF].

Let us present a brief outline of the proof, say for a variety X with pseudoeffective tangent sheaf.

The fact that T X is pseudoeffective pullbacks and restricts to one factor in the Beauville-Bogomolov decomposition of X. Supposing by contradiction that X has no abelian part, we can reduce to a Calabi-Yau or IHS factor Z such that T Z is pseudoeffective. The work of [START_REF] Greb | Klt varieties with trivial canonical class. Holonomy, differential forms and fundamental groups[END_REF] also establishes that T Z and all its symmetric powers are stable of slope zero with respect to any polarisation H. Finally, since Z is not abelian, its orbifold second Chern class satisfies ĉ2 (T Z )•H dim X-2 ̸ = 0. This contradicts the following generalization of [86, Theorem 1.1]: Theorem 3.3. Let X be a normal projective variety with klt singularities of dimension n, H a Q-Cartier ample divisor on X. Consider E a reflexive sheaf on X such that:

• ĉ1 (E) • H n-1 = 0; • for some l ≥ 6, S [l] E is H-stable; • E is pseudoeffective. Then ĉ1 (E) 2 • H n-2 = ĉ2 (E) • H n-2 = 0.
Moreover, there is a finite Galois covering ν : X → X, étale in codimension 1, such that ν [ * ] E is locally-free, has a numerically trivial determinant, and is Gal( X/X)equivariantly flat on X, ie comes from a Gal( X/X)-equivariant representation of π 1 ( X). In particular, ν [ * ] E is numerically flat, and, as symmetric multilinear forms on NS(X):

c 1 (ν [ * ] E) ≡ 0, c 2 (ν [ * ] E) ≡ 0.
The hard part here is the first assertion on the vanishing of orbifold Chern classes, the rest follows from [START_REF] Lu | A characterization of finite quotients of abelian varieties[END_REF].

In Sections 2.3, 2.4, 2.5, we recalled and proved basics to reduce the proof of Theorem 3.3 to working on a normal projective klt surface S. A crucial ingredient is that orbifold Chern classes behave well under certain restrictions [START_REF] Greb | The Miyaoka-Yau inequality and uniformisation of canonical models[END_REF]Proposition 3.11]. In Section 2.6, we introduced an unfolding p : Ŝ → S, obtained by gluing together local finite Galois quasiétale resolutions of the singularities of S. The surface Ŝ may be as singular as S; importantly enough though, any reflexive sheaf E on S reflexively pulls back to a locally-free sheaf Ê on Ŝ. We investigate the relationship of E and Ê. The key of the proof of Theorem 3.3, presented in Chpater 4, is then to establish the nefness of Ê, which yields the Chern classes vanishing for Ê, hence for E.

Note that E may very well not be nef itself: see Remark 2.25.

As a conclusive remark, note that investigating pseudoeffectivity of the tangent and reflexivized cotangent sheaves of a variety with trivial canonical class requires knowledge of its singular Beauville-Bogomolov decomposition. To that extent, Theorem 3.2 cannot be used on an explicit variety before knowing a bare minimum about its geometry. In Section 6, we exhibit 2409 Calabi-Yau threefolds with singularities in codimension 2, among the 7555 wellformed quasismooth hypersurfaces of trivial canonical sheaf in weighted projective 4-dimensional spaces classified by [START_REF] Kreuzer | No mirror symmetry in Landau-Ginzburg spectra![END_REF][START_REF] Kreuzer | LG/cft: LG models, orbifolds, and weighted projective space[END_REF]. These examples stay out of the range of the earlier pseudoeffectivity result of [START_REF] Höring | Algebraic integrability of foliations with numerically trivial canonical bundle[END_REF]Theorem 1.6], but are covered by our Theorem 3.2.

CHAPTER 4

RESTRICTING TO A GENERAL SURFACE

We prove the following proposition in Section 4.2: Proposition 4.1. Let S be a normal projective klt surface, and H an ample Q-Cartier divisor on S. Let E be a reflexive sheaf on S such that:

• ĉ1 (E) • H = 0;
• for some l ≥ 6, S [l] E is stable with respect to H;

• E is pseudoeffective.
Then there is an unfolding p : Ŝ → S as in Section 2.6 on which the locally-free sheaf

Ê = p [ * ] E is nef.
In Section 4.1, we explain how this result implies the first part of Theorem 3.3, namely the vanishing of the squared first and second orbifold Chern classes. Lemma 4.2. Let S be a normal projective surface, H an ample Q-Cartier divisor on S and E a locally-free sheaf on S. Assume that E is nef and c 1 (E) • H = 0. Then:

c 1 (E) 2 = c 2 (E) = 0. Proof. Let S ε
→ S be the minimal resolution of S, H = ε * H. Writing Ẽ := ε * E, we get a nef locally-free sheaf on a smooth surface. The functoriality of Chern classes of locally-free sheaves by continuous pullbacks [START_REF] Milnor | Characteristic classes[END_REF]

, XI-Lemma 1] guarantees c i ( Ẽ) = ε * c i (E) for i = 1, 2. In particular, c 1 ( Ẽ) • H = 0. By nefness, c 1 ( Ẽ) 2 ≥ 0. Hence, by Hodge Index Theorem, c 1 ( Ẽ) 2 = 0 which yields, by [47, Proposition 2.1, Theorem 2.5], c 2 ( Ẽ) = 0. So we obtain: c 1 (E) 2 = c 2 (E) = 0.
Proof of the first assertion in Theorem 3.3. Let a variety X, an ample Q-Cartier divisor H, and a reflexive sheaf E be as in the asssumptions of Theorem 3. • as a consequence, S [l] (E| S ) = (S [l] E)| S ;

• S [l] (E| S ) remains H| S -stable of zero slope;

• E| S is pseudoeffective.

Then, by Proposition 4.1, there is a finite Galois cover p : Ŝ → S such that the reflexive pullback Ê := p [ * ] E| S is a nef locally-free sheaf of zero slope. Lemma 4.2 yields:

c 1 ( Ê) 2 = c 2 ( Ê) = 0,
so that, by construction, ĉ1 2 (E| S ) = ĉ2 (E| S ) = 0 and hence:

ĉ1 2 (E) • H n-2 = ĉ2 (E) • H n-2 = 0.
The first assertion in Theorem 3.3 is established.

Proof of Proposition 4.1.

Let S be a normal projective klt surface, and H an ample Q-Cartier divisor on S. Let E be a reflexive sheaf on S such that:

• ĉ1 (E) • H = 0;
• for some l ≥ 6, S [l] E is stable with respect to H;

• E is pseudoeffective.

As in Section 2.6, we denote by p : Ŝ → S a finite Galois cover on which the sheaf Ê = p [ * ] E is locally-free. Let Ĥ := p * H be an ample Q-Cartier divisor on Ŝ, π : P( Ê) → Ŝ be the natural map and ζ be the tautological bundle on P( Ê).

Abiding by [START_REF] Höring | Algebraic integrability of foliations with numerically trivial canonical bundle[END_REF]Sect.3.2], we prove two lemmas. The first lemma uses the stability of S [l] E to prove the ampleness of ζ on certain subvarieties of P( Ê). Proof. If Z is contained in a fiber of π, then Z ∩ π-1 ( Ĉ) is empty and there is nothing to prove. Assume now that π(Z) has dimension 1 or 2 in Ŝ. Since p is finite, p(π(Z)) has dimension 1 or 2 in S. Hence, for m big and divisible enough, a very general curve C ∈ |mH| satisfies:

• C is a smooth curve inside the locus S 0 ⊂ S reg where E is locally-free;

• Ĉ := p -1 (C) is general in the basepoint-free linear system p * |mH| and hence smooth too;

• consequently, we have locally-free sheaf isomorphisms Ê|

Ĉ = p * E| C and S l (E| C ) = (S [l] E)| C ;
• since mH is ample and π(Z) is not a point, Z ∩ π-1 ( Ĉ) is non empty;

• since Z is proper in P(E), Z ∩ π-1 ( Ĉ) is proper in π-1 ( Ĉ);
⊂ π-1 ( Ĉ) = P( Ê| Ĉ ), Ä ζ| π-1 ( Ĉ) ä dim W • W > 0.
Using this formula for any closed subvariety W of Z ∩ π-1 ( Ĉ), the Nakai-Moishezon criterion shows that ζ| Z∩π -1 ( Ĉ) is ample.

The second lemma is set at the higher level of ( Ŝ, Ê) directly. It uses the pseudoeffectivity and Ĥ-semistability of the locally-free sheaf Ê, infered by Proposition 2. [START_REF] Chen | Birational boundedness of rationally connected Calabi-Yau 3-folds[END_REF]

nef. Let Z ⊂ B -( ζ) be an irreducible component of maximal dimension. Since ζ is pseudoef- fective, Z is proper in P( Ê). Note that Z contains a ζ-negative curve N : its image µ ′ (N ) must be a ζ-negative curve, hence it is not in a fiber of π. So π(µ ′ (Z)) is not a point in Ŝ.
Now, for m big and divisible enough, for a general curve Ĉ ∈ p * |mH|,

• Ĉ is a smooth curve in Ŝreg ; in particular, µ is an isomorphism over Ĉ;

• Z ∩ µ ′-1 (π -1 ( Ĉ)) is non-empty, and proper in µ ′-1 (π -1 ( Ĉ));

• Ê| Ĉ is nef by Lemma 2.36 and Proposition 2.39 and it has Ĥ| Ĉ -slope zero;

• hence, ζ| µ ′-1 (π -1 ( Ĉ)) is nef too, and moreover its top power is zero;

• hence, by [START_REF] Höring | Algebraic integrability of foliations with numerically trivial canonical bundle[END_REF]Lemma 3.4] (which applies with a void condition for k = codim P( Ê) Z, since Z was chosen with maximal dimension, i.e., with minimal codimension):

0 = Ä ζ| µ ′-1 (π -1 ( Ĉ)) ä dim µ ′-1 (π -1 ( Ĉ)) ≥ Ä ζ| Z∩µ ′-1 (π -1 ( Ĉ)) ä dim Z∩µ ′-1 (π -1 ( Ĉ)) ≥ 0.
As µ ′ is an isomorphism over Ĉ, W := µ ′ (Z) works well as the closed proper subvariety of P( Ê) we want to construct.

We now combine these lemmas to establish Proposition 4.1.

Proof of Proposition 4.1. Suppose by contradiction that Ê is not nef. Then Lemma 4.4 yields a closed proper subvariety W of P( Ê) which satisfies, for m big and divisible enough and for a very general curve Ĉ ∈ p * |mH|:

∅ ̸ = W ∩ π-1 ( Ĉ) ⊊ π-1 ( Ĉ) and ζ| W ∩π -1 ( Ĉ) is nef
• E is H-semistable;
• the following equalities hold:

ĉ1 (E) • H n-1 = ĉ1 2 (E) • H n-2 = ĉ2 (E) • H n-2 = 0.
Then there is a finite Galois morphism ν : X → X, étale in codimension 1, such that ν [ * ] E is a locally-free sheaf with numerically trivial determinant, and is Gal( X/X)equivariantly flat. Consequentially, ν [ * ] E is numerically flat and its first and second Chern classes are numerically trivial.

Proof. We apply [START_REF] Lu | A characterization of finite quotients of abelian varieties[END_REF]Theorem 1.4] to obtain a finite Galois morphism ν : X → X, étale over X reg , such that ν [ * ] E is locally-free with a numerically trivial determinant and Gal( X/X)-equivariantly flat. Let then ε : X′ → X be a resolution of X and E ′ := ε * ν [ * ] E, which is a flat locallyfree sheaf with a numerically trivial determinant on X′ . As shown in [START_REF] Höring | Algebraic integrability of foliations with numerically trivial canonical bundle[END_REF]Rmk.2.6], E ′ is then numerically flat and its Chern classes vanish (as cohomological classes on X′ ). By Prop.2.23,

ν [ * ] E is nef, hence numerically flat. Moreover, for any Q-Cartier divisors D 1 , . . . , D n-2 , c 2 (ν [ * ] E) • D 1 • • • D n-2 = c 2 (E ′ ) • ε * D 1 • • • ε * D n-2 = 0,
so the Chern classes of ν [ * ] E are trivial, which completes the proof of the theorem.

Proof of Theorem 3.2.

We give a few definitions along the lines of Theorem 3.1: Definition 5.2. Let X be a normal projective canonical variety of dimension n ≥ 2. It is called:

• a Calabi-Yau variety if h 0 (Y, Ω [q]
Y ) = 0 for all integers 1 ≤ q ≤ n -1 and all quasiétale finite covers Y → X;

• an irreducible holomorphic symplectic (IHS) variety if there is a reflexive form σ ∈ H 0 (X, Ω

X ) such that, for any quasiétale finite cover f : Y → X, the reflexive form f

[ * ] σ generates H 0 (Y, Ω [•]
Y ) as an algebra for the wedge product. We use the terms singular Calabi-Yau (resp. IHS) variety and Calabi-Yau (resp. IHS) variety interchangeably, unless explicitly said otherwise. They may both accidentally denote smooth varieties. Definition 5.3. For the sake of a consistent terminology, let us call a singular K3 surface, or for short a K3 surface, a normal projective klt surface which has no finite quasiétale cover by an abelian variety. Equivalently, it is a Calabi-Yau variety or an IHS variety of dimension 2. Definition 5.4. For the sake of a convenient vocabulary, let us define the augmented irregularity q(X) of a normal projective klt variety X with trivial canonical class as the maximum of all irregularities q(Y ) of finite quasiétale covers Y of X. Note that it is precisely the dimension of the abelian part in the singular Beauville-Bogomolov decomposition of X.

Let us now proceed to prove Theorem 3.2.

Proof of Theorem 3.2. Let X be a normal projective klt variety of dimension at least 2 with trivial canonical class. Suppose that Ω [1] X is pseudoeffective (the same whole argument works just alike for the tangent sheaf T X ) and assume by contradiction that q(X) = 0.

The singular Beauville-Bogomolov decomposition then reads:

f : X → X and X ∼ = i Y i × j Z j ,
with the same notations as in Theorem 3.1.

Remember that f [ * ] Ω [1] X = Ω [1]
X , since reflexive sheaves are normal and there is a big open set over which f is just a finite étale cover. By Proposition 2.33, Ω [1] X is pseudoeffective; it splits according to the product defining X. So there is a factor Y (Calabi-Yau or IHS) of X such that Ω [1] Y is pseudoeffective [86, inductive argument in Proof of Theorem 1.6]. Now, Ω [1] Y satisfies all hypotheses of Theorem 3.3, the stability assumptions coming from [START_REF] Greb | Singular spaces with trivial canonical class[END_REF]Prop.8.20] and [START_REF] Greb | Klt varieties with trivial canonical class. Holonomy, differential forms and fundamental groups[END_REF]Rmk.8.3].

As a consequence, for some ample polarization H on Y , ĉ2 (Ω

[1] Y ) • H dim Y = 0, so that Y
m ∈ N, h 0 (X, S [m] Ω [1] X ) = 0.
Examples for Theorem 3.2 are to search among normal projective klt varieties with trivial canonical class singularities in codimension 2, which are plethoric. But singular varieties whose decomposition is known are not so numerous; and, for sure, one shall understand the Beauville-Bogomolov type of a given variety before telling anything about the positivity of its reflexivized cotangent sheaf.

Example 5.6. A first example to which Theorem 3.2 applies is the following [70, Par.14.2.2]: let F be a Fano manifold on which a finite group G acts freely in codimension 1. Suppose there is a smooth G-invariant element Y in the linear system | -K F |. Then, Y is a smooth Calabi-Yau variety with a G-action. If the volume form on Y is preserved by this action, then X := Y /G is a normal projective klt variety with trivial canonical class, and the morphism Y → X has no ramification divisor, hence it is quasiétale. The fact that the decomposition of X consists of a smooth Calabi-Yau manifold Y guarantees that X is a singular Calabi-Yau variety, as presented in Definition 5.2.

Although X may well have singularities in codimension 2, they merely stem from its global quasiétale quotient structure. In particular, [86, Theorem 1.6] actually proves the non-pseudoeffectiveness of T X and Ω [1] X , namely because it applies to Y and converts onto X through Proposition 2.33. Hence, the example is quite shallow: it has no real need for the machinery dealing with singularities in codimension 2 that Theorem 3.2 is about.

In the next chapter, we present better examples for Theorem 3.2, namely Calabi-Yau threefolds with singularities in codimension 2 that are not constructed as global quasiétale quotients of varieties which are smooth in codimension 2. In Section 5.2, we defined singular Calabi-Yau and IHS varieties. It follows from basic linear algebra that IHS varieties must have even dimension. In particular, the singular Beauville-Bogomolov decomposition for a normal projective klt variety X of dimension 3 is quite simple: X has to be one of the following:

• a smooth abelian variety;

• a product S × E, where S is a K3 surface as in Definition 5.3 and E is a smooth elliptic curve;

• a Calabi-Yau variety.

The aforementioned [START_REF] Lu | A characterization of finite quotients of abelian varieties[END_REF]Theorem 1.4] provides a criterion for identifying the purely abelian case by computing ĉ2 (X).

One is then left with two cases: the singular threefold X may arise from a product S × E, in which case T X and Ω [1] X are pseudoeffective because of the abelian factor E; alternatively, X can be a genuine singular Calabi-Yau threefold. This second possibility is hard to identify, but, when it happens, it may give new examples for Theorem 3.2.

The next subsection is devoted to providing a necessary condition for a normal projective klt threefold to be finitely quasiétaly covered by a product S × E.

6.1 Products of a K3 surface and an elliptic curve. We are going to prove the following result: Proposition 6.1. Let X be a normal projective klt threefold with trivial canonical class. Suppose its Beauville-Bogomolov decomposition is of the form

X = S × E,
where S is a K3 surface and E a smooth elliptic curve. Then X has fibrations:

X / / S/G S E/G E
where G E and G S are finite subgroups of Aut(E) and Aut(S). In particular, ρ(X) ≥ 2.

Let us first state a weak uniqueness result, guaranteeing that the statement of Proposition 6.1 makes sense. It is straightforward from the proof of the Beauville-Bogomolov decomposition theorem. Proposition 6.2. Let X be a normal projective klt variety with trivial canonical class. Then the number, types and dimensions of the factors of a finite quasiétale covering X → X as in Theorem 3.1 do not depend on the choice of that covering.

A finite quasiétale morphism is not necessarily a quotient map by a finite group action free in codimension 1. In the smooth case however, [10, Lemma p.9] allows us to assume that the finite étale decomposition morphism p : X → X is Galois. Let us state a partial singular analog: Proposition 6.3. Let X be a normal projective klt variety with trivial canonical class. Take a finite quasiétale covering f : X → X as in Theorem 3.1. Suppose that all Calabi-Yau factors of X have even dimension. Then there is a finite quasiétale Galois morphism f ′ : Z → X, so that Z splits into factors in the same number, types, and dimensions as X.

Proof. By [72, Theorem 1.5], we can take a finite quasiétale Galois covering g : Y → X such that any finite morphism Z → Y étale over Y reg is étale over Y . By purity of the branch locus, any quasiétale morphism Z → Y is then étale.

Note that Y is still a normal projective klt variety with trivial canonical class, hence has a singular Beauville-Bogomolov decomposition h : Z → Y . By Proposition 6.2, the factors of Z have the same type as those of X. It writes:

Z = A × i Y i × j Z j ,
where A is an abelian variety, Y i Calabi-Yau varieties and Z j IHS varieties. Since all Y i and, of course, all Z j have even dimension, by [START_REF] Greb | Klt varieties with trivial canonical class. Holonomy, differential forms and fundamental groups[END_REF]Cor.13.3], they are simply connected.

Hence, finite étale fundamental groups equal: " π 1 (Z) ≃ " π 1 (A). That is to say, any finite étale cover of Z actually stems from a finite étale cover of A.

We now use [START_REF] Greb | Etale fundamental groups of Kawamata-log-terminal spaces, falt sheaves and quotients of abelian varieties[END_REF]Theorem 3.16]: there is a finite Galois morphism γ : Z → Z such that Γ = g • h • γ : Z → X is finite Galois and ramifies where g • h does. So Γ is still quasiétale, in particular h • γ : Z → Y is quasiétale too. By construction of Y , h • γ is then étale, so that γ is étale. By construction of Z, one has:

Z = A ′ × i Y i × j Z j ,
where A ′ is a finite étale cover of the abelian variety A. Finally, Γ : Z → X is finite Galois quasiétale, and Z splits as mandated.

Remark 6.4. The main obstacle for generalizing this proposition is the fact that fundamental groups of odd-dimensional Calabi-Yau varieties are poorly understood [START_REF] Greb | Klt varieties with trivial canonical class. Holonomy, differential forms and fundamental groups[END_REF]Sect.13.2]; most notably, they may not be finite.

Here is the last ingredient for the proof of Proposition 6.1: Lemma 6.5. Let S be a K3 surface as in Definition 5.3, E a smooth elliptic curve. Then:

Aut(S × E) ∼ = Aut(S) × Aut(E).
Proof. Let S be the minimal resolution of S. It is a smooth K3 surface, so Aut( S) is discrete. Moreover, the uniqueness of minimal resolution implies that any automorphism of S lifts to an automorphism of S, and this is obviously an injection. Hence, Aut(S) is discrete.

Let us now copy the argument by [START_REF] Beauville | Some remarks on Kähler manifolds with c 1 = 0[END_REF]Lemma p.8]. Let u ∈ Aut(S × E). Since the projection p E : S × E → E is the Albanese map of S × E, we can factor p E • u by it:

there is v ∈ Aut(E) such that p E • u = v • p E .
Hence, there is a map w : E → Aut(S) which decomposes: u : (s, e) ∈ S × E → (w e (s), v(e)).

Since Aut(S) is discrete, the map w is constant, so u = (w 0 , v).

Proof of Proposition 6.1. Let X be a normal projective variety of dimension 3 with trivial canonical class. Suppose that there is a finite quasiétale cover f :

S × E → X,
where S is a singular K3 surface and E a smooth elliptic curve. By Proposition 6.3, we can assume that there is a finite group G acting on S × E such that f is the induced quotient map. By Lemma 6.5, G can be considered a subgroup of Aut(S) × Aut(E).

As it acts diagonally, we have the following diagram:

S × E p S / / p E f % % S $ $ E $ $ X / / S/G S E/G E so that ρ(X) is at least 2.
6.2 Calabi-Yau hypersurfaces in weighted projective spaces. The aim of this last part is to provide examples of Calabi-Yau threefolds that are singular along curves, by establishing the following result. Proposition 6.6. Let P = P(w 0 , . . . , w 4 ) be a weighted projective space and d = w 0 + . . . + w 4 such that there is a general wellformed quasismooth hypersurface X of degree d in P. Suppose that X contains no edge of P. Then X is a singular Calabi-Yau in the sense of Definition 5.2.

A general exposition to complete intersections in weighted projective spaces can be found in [START_REF] Iano-Fletcher | Working with weighted complete intersections[END_REF]. We stick to its terminology.

Let P = P(w 0 , . . . , w 4 ) be a wellformed 4-dimensional weighted projective space. There is a ramified quotient map: p : P n → P, by the finite diagonal group action of i Z w i on P n . With homogeneous coordinates on either side, we can write:

p : [x 0 : . . . : x n ] ∈ P n → [y 0 = x w 0 0 : . . . : y n = x wn n ] ∈ P.
We denote by O P (1) the ample Q-Cartier divisor on P whose pullback by p is O P n (1). If the linear system |O P (d)| contains a wellformed quasismooth hypersurface, it actually contains a Zariski-open set of such hypersurfaces and we write X d for a general one.

Singularities of general quasismooth hypersurfaces of dimension 3.

Let X be a general quasismooth hypersurface of degree d and of dimension 3 in the weighted projective space P. Then X is a full suborbifold of P (see [START_REF] Brozellino | Elementary orbifold differential topology[END_REF]Def.5] for a definition, [53, Theorem 3.1.6] for a proof). In particular, X sing = X ∩ P sing , and at any point x ∈ X ∩ P sing , writing that P is locally isomorphic to a quotient C 4 /G x , X is locally isomorphic to C 3 /G x in a compatible way with inclusions. Hence, X has only quotient singularities, so it is klt. The locus X sing is a finite union of curves and points, which may be of various types:

• a vertex in P is a point with y i = 1 for a single i ∈ [[0, 4]] and y j = 0 for all j ̸ = i.

If • a 2-face in P is a 2-plane with equation y j = 0 for all j ∈ J, for a certain J ⊂ [[0, 4]] of cardinal 2. If gcd(w j ) j̸ ∈J ̸ = 1, the 2-face is in P sing . By quasismoothness, no 2face lies entirely in X. Hence, any 2-face intersects X along an effective 1-cycle. In this way, 2-faces in P sing may produce curves in X sing .

w i ̸ = 1,
Under the additional hypothesis that X contains no edge of P, we can say more about singular loci.

Indeed, it is worth noticing that the restricted quotient map p -1 (X) → X is an unfolding of X, as defined in Section 2.6; we may write X for p -1 (X). For establishing Prop.6.6, we will prove the following: Lemma 6.7. Let X be a general wellformed quasismooth hypersurface of dimension 3 in a weighted projective space P not isomorphic to P 4 . Assume that X has trivial canonical class and that it contains no edge of P. Then ĉ2 (X) • O X (1) > 0.

In the course of the proof of this lemma, we will use the fact that X containing no edge of P, X is smooth in codimension 2. Remark 6.8. Note that the restricted finite map X = p -1 (X) → X is certainly ramified along divisors, so that X, in the lucky case where it happens to be a singular Calabi-Yau threefold, is not at all constructed as a finite quasiétale global quotient, contrarily to the unsatisfying Example 5.6.

The proof that X is smooth in codimension 2 relies on the following lemma and remark: Lemma 6.9. Let X be a general quasismooth hypersurface of degree d in the weighted projective space P = P(w 0 , . . . , w 4 ). Suppose that it contains no edge of P. Then the base locus Bs(O P (d)) has dimension 0.

Proof. Let Z be an irreducible component of the base locus of O P (d), let us prove by induction on dim P that it is a point. Suppose we are at the induction step where the ambient space P ′ has local coordinates y 0 , y 1 , y 2 , . . . and dimension 4, 3 or 2.

Denote by H i the hyperplane {y i = 0} in P ′ , by P ′ i the isomorphic weighted projective space P ′ (. . . , ŵi , . . .). By [14, Prop.4.A.3], we have an isomorphism between the restriction O P ′ (d) ⊗ O H i and the Q-Cartier divisor O P ′ i (d). This translates to global sections as a surjection:

H 0 (P ′ , O P ′ (d)) ↠ H 0 (P ′ i , O P ′ i (d)), (6.1)
which is given by setting y i = 0 when considering the global sections as certain polonomials in the local coordinates of P ′ . The quasismoothness of X in P and the way the composite surjection

H 0 (P, O P (d)) ↠ H 0 (P ′ , O P ′ (d)),
writes in local coordinates yield a global section of O P ′ (d) of the form y α 0 0 y α 1 1 y α 2 2 . In particular, there is an i = 0, 1 or 2 such that Z ⊂ H i ≃ P ′ i . Moreover, by Eq.6.1, Z sits in the base locus of O P ′ i (d). Induction propagates from P ′ = P down to when we obtain that Z is contained in an edge H ijk of P and in the base locus Bs(O P ijk (d)) ⊂ Bs(O P (d)) ⊂ X. Since X contains no edge of P, Z is in X ∩ H ijk of dimension 0, so it is a point. Remark 6.10. With the same notations and hypotheses, the intersection of X with any 2-face of P sing is a reduced curve.

Proof. As in the proof of Lemma 6.9, the intersection is scheme-theoretically defined by a general section of O P ij (d). We are to show that such general section of O P ij (d) is quasismooth in the weighted projective space P ij , hence it is a variety by [89, 3.1.6].

We use the arithmetical criterion for quasismoothness: since X contains no edge of P, each pair w a , w b partitions d. We are left to check the criterion for k = 1: fix any a ̸ = i, j, we want to find b ̸ = i, j such that w a divides d -w b . It is clear that there is a b ∈ [[0, 4]] satisfying that. As H ij is a 2-face in P sing , the greatest common divisor of all weights except w i , w j is non-trivial, divides d but neither w i nor w j (by wellformedness). In particular, since this greatest common divisor divides

w b = d -αw a , b ̸ = i, j, as wished.
We can now deduce: Proposition 6.11. Let X be a general quasismooth hypersurface of degree d in a weighted projective space P = P(w 0 , . . . , w 4 ), p the natural quotient P 4 → P, X = p -1 (X). Suppose that X contains no edge of P. Then X is smooth in codimension 2.

Proof. The threefold X is general in the linear system p * |O P (d)|, whose base locus has dimension 0 by Lemma 6.9. By Bertini's theorem, X is smooth in codimension 2. Remark 6.12. The converse of Proposition 6.11 does not hold: for instance, the general quasismooth X 7 in P(1, 1, 1, 2, 2) contains the edge of equation y 0 = y 1 = y 2 = 0, but its unfolding is nevertheless smooth in codimension 2.

Example 6.13. The hypothesis of Proposition 6.11 is not that X contains no edge of P sing , but that it contains no edge of P at all: for instance, consider the general X = X 56 in P [START_REF] Amerik | Morrison-Kawamata cone conjecture for hyperkähler manifolds[END_REF][START_REF] Andreatta | Special rays in the Mori cone of a projective variety[END_REF][START_REF] Balaji | Holonomy groups of stable vector bundles[END_REF][START_REF] Bellamy | All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces[END_REF][START_REF] Campana | Projective manifolds whose tangent bundles are numerically effective[END_REF]. It contains a single edge of P, namely e of equation y 0 = y 1 = y 4 = 0. This edge does not actually lie in P sing , as 9 and 13 are coprime, but one can check that X has the curve p -1 (e) in its singular locus (by computing the derivatives of the equation defining X in P 4 along the curve p -1 (e)).

Example 6.14. The general wellformed quasismooth hypersurface X = X 1734 in P(91, 96, 102, 578, 867) contains no edge of P. In particular, X is smooth in codimension 2 by Proposition 6.11.

Moreover, the curves of X sing are precisely the intersections of X with all 2-faces of P sing , which we can list:

• y 0 = y 1 = 0 of type 1 17 [START_REF] Auffarth | Smooth quotients of complex tori by finite groups[END_REF][START_REF] Beauville | Variétés kählériennes dont la première classe de Chern est nulle[END_REF],

• y 0 = y 3 = 0 of type 1 3 (1, 2), • y 0 = y 4 = 0 of type 1 2 (1, 1
). It is possible to check the type of singularities of a general hypersurface of a given degree in a given weighted projective space by a simple computer program.

Proof of Proposition 6.6. As we said before, the main ingredient in the proof is Lemma 6.7.

Proof of Lemma 6.7. Let p : P 4 → P be the natural quotient map. Writing P = P(w 0 , . . . , w 4 ) with (w 0 , . . . , w 4 ) not colinear to (1, . . . , 1), the morphism p has degree w 0 • • • w 4 , which we denote by N , and X has degree w 0 + . . . + w 4 , which we denote by d. We may also write s for the symmmetric elementary polynomial of degree 2 in the weights and q for the sum of their squares:

d 2 = q + 2s.
Since X is a full suborbifold of P, X := p -1 (X) → X is an unfolding of X as defined in Section 2.6. Applying the left-exact functor of reflexive pullback (see Lemma 2.15) to the exact sequence:

0 → T X → T P | X → -K P ,
we get another exact sequence:

0 → p [ * ] T X → p [ * ] T P | X → p [ * ] (-K P ) → Z → 0,
where the coherent sheaf Z is supported on the locus p -1 (Sing X) ⊂ X of codimension at least 2.

Because of the last surjection, dim k(p) Z ⊗ O p ≤ 1 for any closed point p ∈ X. By Proposition 6.11, the unfolding X is smooth in codimension 2, so the usual second Chern class c 2 (Z) makes sense. Since usual Chern classes are additive, and

c 1 (T X ) = 0, c 1 (Z) = 0: ĉ2 (T X ) • O X (1) = ĉ2 (T P | X ) • O X (1) + 1 N c 2 (Z) • O X (1)
.

By the Miyaoka-Yau inequality [76, Theorem 1.5], we have a positive contribution:

ĉ2 (T P | X ) • O X (1) = ĉ2 (T P ) • (-K P ) • O P (1) ≥ 4 10 (-K P ) 3 • O P (1) = 4d 3 10N .
Let us estimate the other summand. Take m big and divisible enough that O X (m) is very ample and S a general element in |O X (m)|. By [106, Lemma 10.9],

c 2 (Z) • O X (1) = 1 m c 2 (Z| S ) = - 1 m deg(Z| S )
Denote by C 1 , . . . C k the curves in X sing . By Lemma 6.15, we can bound:

deg(Z| S ) ≤ Card Ç S ∩ k i=1 p -1 (C i ) å = k i=1 N O X (m) • C i ≤ N mO X (1) 3 0≤i<j≤4 w i w j = mN s(-K P ) • O P (1) 3 = msd.
Finally putting the positive and negative part together,

ĉ2 (X) • O X (1) > 4d 3 -10sd 10N = d(4q -2s) 10N = d 10N 0≤i<j≤4 (w i -w j ) 2 > 0.
Lemma 6.15. Let X be a general wellformed quasismooth hypersurface of dimension 3 in a weighted projective space P. Assume that X has trivial canonical class and contains no edge of P. Then there are at most 10 curves in X sing , with different cohomological classes in the list of the

[O X (w i ) • O X (w j )] ∈ H 4 (X; Q), for 0 ≤ i < j ≤ 4.
Proof of Lemma 6.15. By Remark 6.10, each curve in X sing is scheme-theoretically the complete intersection of X with a 2-face H ij of P sing . This association being bijective, there are as many curves in X sing as 2-faces in P sing , so at most 10. The curve that corresponds to the 2-face

H ij has cohomological class [O X (w i ) • O X (w j )].
Now we can finally establish Proposition 6.6:

Proof. Consider X a general wellformed quasismooth hypersurface of degree d = w 0 + . . . + w 4 in a weighted projective space P = P(w 0 , . . . , w 4 ). Suppose that X contains no edge of P. If P is P 4 , X is smooth and there is nothing to prove. Let us assume P ̸ ∼ = P 4 . By Lemma 6.7, ĉ2 (X) • O X (1) ̸ = 0, hence by [133, Theorem 1.4], X is not a finite quotient of an abelian threefold. Moreover, one has Pic(X) ≃ Z [53, Theorem 3.2.4(i)], so Proposition 6.1 applies to X: it is not covered by a product of a K3 surface and an elliptic curve, hence its Beauville-Bogomolov decomposition consists of a single Calabi-Yau factor. By Lemmma 6.16, so X itself is a Calabi-Yau variety, in the sense of Definition 5.2. In particular, X has canonical (and not merely klt) singularities. Lemma 6.16. Let X be a general quasismooth hypersurface in a weighted projective space P. Then any finite quasiétale cover X ′ of X is trivial.

Proof. Let X ′ be a finite quasiétale cover of X of degree d; note that by Zariski purity of branch locus, it is étale over X reg . Let C * X ⊂ C n+1 \ {0} be the smooth cone over X, with the projection q : C * X → X. The morphism

C ′ = X ′ × X C * X → C * X is
(C * X ) = {1} so d = 1.
Examples for Proposition 6.6. General wellformed quasismooth hypersurfaces with trivial canonical class in 4-dimensional weighted projective spaces are classified in [START_REF] Kreuzer | No mirror symmetry in Landau-Ginzburg spectra![END_REF]. There is an explicit exhaustive list of the 7555 of them. In this list, 7238 elements are not smooth in codimension 2, and 2409 elements that are not smooth in codimension 2 also contain no edge of their ambient weighted projective space. These elements fulfill the hypotheses for Proposition 6.6, just as Example 6.14 did: so they are singular Calabi-Yau threefolds to which Theorem 3.2 applies. The exhaustive enumerations of elements of the [START_REF] Kreuzer | No mirror symmetry in Landau-Ginzburg spectra![END_REF] classification satisfying additional properties were done by running a simple computer program on the database [START_REF] Kreuzer | LG/cft: LG models, orbifolds, and weighted projective space[END_REF]. The program is available on my webpage https://math.unice.fr/~gachet/ research.html. Remark 6.17. For the sake of transparent terminology, let us explain why the varieties studied in [START_REF] Kreuzer | No mirror symmetry in Landau-Ginzburg spectra![END_REF] are the same as general quasismooth wellformed hypersurfaces of trivial canonical class in a 4-dimensional weighted projective space.

First, any variety that [START_REF] Kreuzer | No mirror symmetry in Landau-Ginzburg spectra![END_REF] calls a nondegenerate Calabi-Yau hypersurface is sitting in an open set of nondegenerate Calabi-Yau hypersurfaces of a given linear system. This is precisely what we refered to as a general quasismooth hypersurface of trivial canonical class in a weighted projective space.

The paper [START_REF] Kreuzer | No mirror symmetry in Landau-Ginzburg spectra![END_REF] classifies " tuples "of positive integers d, {{w 0 , . . . , w N }} such that:

• there are no " trivial variables ", ie d 2 ̸ ∈ {{w 0 , . . . , w N }},

• N = 3 or 4,
• there is a nondegenerate Calabi-Yau hypersurface of degree d in P(w 0 , . . . , w N ) with condition " c = 9 ".

Here, we use {{•}} to denote tuples where order does not matter, or equivalently sets where elements may appear with a certain multiplicity. We claim that the map f :

d, {{w 0 , . . . , w N }} → ß d, {{w 0 , . . . , w 3 , d 2 }} if N = 3 d, {{w 0 , . . . , w 4 }}
else, is a one-to-one correspondence between the data of [START_REF] Kreuzer | No mirror symmetry in Landau-Ginzburg spectra![END_REF] and all tuples d, {{w 0 , . . . , w 4 }} such that X d ⊂ P(w 0 , w 1 , w 2 , w 3 , w 4 ) is a general quasismooth wellformed hypersurface with trivial canonical class. To prove this claim, let us compute the image of this injective map f : a tuple d, {{w 0 , . . . , w 4 }} is in the image of f if and only if at most one of the w i equals d 2 and the general hypersurface of degree d in the projective space of weights {{w i | 2w i ̸ = d}} is quasismooth, has trivial canonical class and satisfies the rewritten condition c = 9:

4 i=0 1 - 2w i d = 3, ie 4
i=0 w i = d.
So, it is clear that the image by f of the [START_REF] Kreuzer | No mirror symmetry in Landau-Ginzburg spectra![END_REF] tuples with N = 4 is made of all d, {{w 0 , . . . , w 4 }} such that X d ⊂ P(w 0 , . . . , w 4 ) is a general quasismooth wellformed hypersurface of trivial canonical class, and for all i, 2w i ̸ = d.

The image by f of the tuples with N = 3 is easily checked to stand for quasismooth hypersurfaces in weighted projective spaces of dimension 4. We check that the quasismooth hypersurfaces arising in that way are wellformed by a careful application of the criterion [89, 6.13], [51, Prop.2], together with elementary arithmetic. Notably, the fact that

INTRODUCTION

Since singularities are a byproduct of the Minimal Model Program, studying singular varieties with trivial canonical class, or singular K-trivial varieties, is an important question in the birational classification of complex algebraic varieties. From this point of view, the recent generalization of the Beauville-Bogomolov decomposition theorem for smooth K-trivial varieties ( [START_REF] Beauville | Variétés kählériennes dont la première classe de Chern est nulle[END_REF]) to klt K-trivial varieties ( [START_REF] Greb | Klt varieties with trivial canonical class. Holonomy, differential forms and fundamental groups[END_REF][START_REF] Druel | A decomposition theorem for singular spaces with trivial canonical class of dimension at most five[END_REF][START_REF] Höring | Algebraic integrability of foliations with numerically trivial canonical bundle[END_REF][START_REF] Bakker | Algebraic approximation and the decomposition theorem for Kähler Calabi-Yau varieties[END_REF]) is highly relevant. It indeed establishes that, after a finite quasiétale cover, any klt Ktrivial variety is a product of a smooth abelian variety, some irreducible holomorphic symplectic varieties with canonical singularities, also called hyperkähler varieties, and some Calabi-Yau varieties with canonical singularities. These three main families of K-trivial varieties are the subject of large, mostly disjoint realms of the literature, ranging from the well-known theory of abelian varieties (exposed notably in the reference books [START_REF] Birkenhake | Complex abelian varieties[END_REF][START_REF] Shimura | Complex multiplication of abelian varieties and its applications to number theory[END_REF]), through the thriving study of hyperkähler varieties (see [START_REF] Debarre | Hyperkähler manifolds[END_REF]1,[START_REF] Kamenova | Survey of finiteness results for hyperkähler manifolds[END_REF] for surveys), to the unruly "Zoo of Calabi-Yau varieties", populated by a huge amount of examples ( [START_REF] Kreuzer | Classification of reflexive polyhedra in three dimensions[END_REF][START_REF] Kreuzer | Complete classification of reflexive polyhedra in four dimensions[END_REF] for K3 surfaces and Calabi-Yau threefolds embedded as hypersurfaces in toric varieties only), and whose boundedness is yet not established (see [START_REF] Wilson | Boundedness questions for Calabi-Yau threefolds[END_REF][START_REF] Wilson | Calabi-Yau threefolds with Picard number three[END_REF][START_REF] Chen | Birational boundedness of rationally connected Calabi-Yau 3-folds[END_REF][START_REF] Di Cerbo | Birational boundedness of low-dimensional elliptic Calabi-Yau varieties with a section[END_REF][START_REF] Birkar | Boundedness of elliptic Calabi-Yau varieties with a rational section[END_REF] for recent breakthroughs).

A new feature appearing in the context of singular K-trivial varieties is that birational morphisms may change the type of the Beauville-Bogomolov decomposition. For example, Kummer surfaces are K3 surfaces, but arise as minimal resolutions of finite quasiétale quotients of abelian surfaces. Similar examples of dimension 3 are numerous, as in [START_REF] Oguiso | On the complete classification of Calabi-Yau threefolds of type III 0[END_REF][START_REF] Oguiso | On certain rigid fibered Calabi-Yau threefolds[END_REF]153], and even less well understood in higher dimensions, cf. [START_REF] Cynk | Higher-dimensional modular Calabi-Yau manifolds[END_REF][START_REF] Cynk | Generalised Kummer constructions and Weil restrictions[END_REF][START_REF] Paranjape | Quotients of E n by A n+1 and Calabi-Yau manifolds[END_REF][START_REF] Andreatta | On the Kummer construction[END_REF][START_REF] Burek | Higher dimensional Calabi-Yau manifolds of Kummer type[END_REF]. In arbitrary dimension, it is known that a crepant resolution or terminalization only changes the type of a klt K-trivial variety if its decomposition entails an abelian factor ([55, Prop.4.10]). 1 This paper aims at describing changes of the type of a K-trivial variety through a birational morphism in the simplest case of higher dimension, i.e., when a singular variety with Beauville-Bogomolov decomposition of purely abelian type is resolved by a Calabi-Yau manifold. We work in the following set-up: By a Calabi-Yau manifold, we mean a smooth simply-connected complex projective variety of dimension n with trivial canonical bundle, without any global holomorphic differential form of degree

i ∈ [[1, n -1]].
Extending the terminology of [START_REF] Oguiso | On algebraic fiber space structures on a Calabi-Yau 3-fold[END_REF], we define n-dimensional Calabi-Yau 1 It reflects the more general fact that the Beauville-Bogomolov decomposition type of a klt Ktrivial variety X with non-trivial fundamental group π 1 (X reg ) is not captured by its algebra of global holomorphic differential forms

H 0 Ä X, Ω X [•]
ä

. Many examples supporting this fact are exposed in [70, Sec.14], and most notably, smooth K-trivial threefolds with Beauville-Bogomolov decomposition of pure abelian type and algebra of global differential forms generated by the volume form (as for a Calabi-Yau threefold) are classified in [START_REF] Oguiso | Calabi-Yau threefolds of quotient type[END_REF].

manifolds of type n 0 as follows.

Theorem 7.1. Theorem 2.119 Let X be a Calabi-Yau manifold of dimension n. The following are equivalent:

(i) There is a nef and big divisor D on X such that c 2 (X) • D n-2 = 0.

(ii) There is an abelian variety A and a finite group G acting freely in codimension 2 on A such that X is a crepant resolution of A/G.

If it satisfies these conditions, X is called a Calabi-Yau manifold of type n 0 .

Calabi-Yau threefolds of type III 0 appear naturally when classifying extremal contractions of Calabi-Yau threefolds [START_REF] Oguiso | On algebraic fiber space structures on a Calabi-Yau 3-fold[END_REF], and fit in a more general circle of ideas on how the cubic intersection form and the second Chern class determine the birational geometry of a Calabi-Yau threefold (see, e.g., the work of Wilson [START_REF] Wilson | The Kähler cone on Calabi-Yau threefolds[END_REF], Oguiso and Peternell [START_REF] Oguiso | Calabi-Yau threefolds with positive second Chern class[END_REF]). Calabi-Yau threefolds of type III 0 were classified by Oguiso, as we now recall. Theorem 7.2. [START_REF] Oguiso | On the complete classification of Calabi-Yau threefolds of type III 0[END_REF] There are exactly two Calabi-Yau threefolds X 3 , X 7 of type III 0 . They are the unique crepant resolution of E j 3 quotiented by the group generated by jid 3 , and of E u 7 3 quotiented by the group generated by:

Ñ 0 -8 7 -10u 7 1 -6 -2u 7 11 -u 7 0 -1 -2u 7 6 + 3u 7 é .
where j = e 2iπ/3 , ζ 7 = e 2iπ/7 ,

u 7 = ζ 7 + ζ 7 2 + ζ 7 4 = -1+i √ 7 2
, and for any complex number z ∈ C \ R, we denote by E z the elliptic curve C/(Z ⊕ zZ).

Our first theorem restricts the isogeny type of A in arbitrary dimension.

Theorem 7.3. Let A be an abelian variety of dimension n and G be a finite group acting freely in codimension

2 on A. If A/G has a crepant resolution that is a Calabi- Yau manifold, then A is isogenous to E j n or to E u 7
n and G is generated by its elements that admit fixed points in A.

Moreover, the local geometry of A/G is generally quite similar to the 3-dimensional model (see Theorem 7.6 below). Two important consequences of this are the following results. Although local arguments are crucial to the proofs of these two results, they are not sufficient to conclude on their own, and we have to resort to global arguments involving the action on the abelian variety in the proofs.

On the one hand, the Calabi-Yau assumption is crucial in Theorem 7.3, as it rules out products of the 3-dimensional examples of Oguiso, e.g., X 3 × X 7 , which is a resolution of a finite quotient of E j

3 × E u 7 3 . On the other hand, Theorem 7.5 merely requires the simply-connectedness of a crepant resolution. Let us explain why. Note that, if A is an abelian variety and G is a finite group acting freely in codimension 2 on A, then A/G cannot have a holomorphic symplectic resolution X. Indeed, a holomorphic symplectic resolution provides (A/G) reg with a holomorphic symplectic form. By [149, Thm, Cor.1] then, since A/G is smooth in codimension 2, it is terminal. As it is Q-factorial as well, it thus admits no crepant resolution. By the Beauville-Bogomolov decomposition theorem, a smooth simply-connected K-trivial fourfold which is not holomorphic symplectic is a Calabi-Yau fourfold, whence the weaker assumption of Theorem 7.6.

The structure of the paper is as follows. Sections 8 to 15.2 build up to the proof of the main technical result.

Theorem 7.6. Let A be an abelian variety of dimension n and G be a finite group acting freely in codimension

2 on A. If A/G has a crepant resolution that is a Calabi- Yau manifold, then (1) A is isogenous to E j n or to E u 7 n
, and G is generated by its elements that admit fixed points in A.

(2) For every translated abelian subvariety W ⊂ A, there is k ∈ N such that the pointwise stabilizer

PStab(W ) := {g ∈ G | ∀w ∈ W, g(w) = w} is isomorphic to Z 3 k if A is isogenous to E j n , or to Z 7 k if A is isogenous to E u 7 n .
(3) For every translated abelian subvariety W ⊂ A, if PStab(W ) is isomorphic to

• Z 3 k
, then there are k generators of it such that their matrices are similar to diag(1 n-3 , j, j, j), and the j-eigenspaces of these matrices are in direct sum.

• Z 7 k , then there are k generators of it such that their matrices are similar to diag( 4 ), and all eigenspaces of these matrices with eigenvalues other than 1 are in direct sum.

1 n-3 , ζ 7 , ζ 7 2 , ζ 7
Our starting point in Section 8 is a necessary condition for a local quotient singularity to have a crepant resolution. The result is the following (Proposition 8.4): If H ⊂ GL n (C) is a finite group, and 0 ∈ U ⊂ C n is an H-stable analytic open set such that U/H admits a crepant resolution, then H is generated by its so-called junior elements, i.e., elements M with eigenvalues (e

2iπa k /d ) 1≤k≤n satisfying 0 ≤ a k ≤ d -1 and a k = d.
Matrices inducing actions on abelian varieties satisfy a rationality requirement [17, 1.2.3], which translates into arithmetic constraints on their characteristic polynomial. These constraints allow us to classify matrices of junior elements g acting on n-dimensional abelian varieties up to similarity: In Section 9, we prove that if a junior element g acts on an abelian variety in a way that the generated group ⟨g⟩ acts freely in codimension 2, then the matrix of g is of one of twelve possible types (see Proposition 9.2). In particular, the order of g and the number of non-trivial eigenvalues of g are bounded independently of the dimension n. The next step is to show that ten out of the twelve types of junior elements can not belong to G, for a mix of local and global reasons. The proof spreads throughout Chapters 10, 11, 13 and 14. Let us sketch the idea of the argument in the simplest case, namely if g is a junior element of composite order other than 6, with at most four non-trivial eigenvalues. If such a junior element g belongs to G, then some non-trivial power g α is not junior, and has a larger fixed locus in A. Fix an irreducible component W of that larger fixed locus that is not in the fixed locus of g: the pointwise stabilizer PStab(W ) ⊂ G does not contain g, but the power g α . Now, as W has codimension less than 4, Chapter 10 shows that PStab(W ) is cyclic generated by one junior element h, and thus, up to possibly replacing h by another junior generator of Fix(W ), one has g α = h α . For well-chosen α, this is enough to yield g = h, and a contradiction.

This idea excludes seven out of the twelve types of junior elements (see Subsection 11.1). The three types of junior elements of order 6 are excluded by technical variations in the next sections. Ruling them out works along with classifying pointwise stabilizers in higher codimension: In codimension 4, Chapter 10 establishes cyclicity of the pointwise stabilizers and Section 11 deduces that junior elements with four non-trivial eigenvalues do not exist; in codimension 5 (Section 13), we first prove that junior elements with five non-trivial eigenvalues do not exist (Subsection 13.1), then deduce cyclicity of the pointwise stabilizers (Subsection 13.2). In codimension 6 (Section 14), we first classify pointwise stabilizers which do not contain junior elements of type diag(1 n-6 , ω, ω, ω, ω, ω, ω): they are isomorphic to Z 3 , Z 7 , Z 3 × Z 3 , Z 7 × Z 7 , or SL 2 (F 3 ) (Subsection 14.1). We use this partial classification to rule out junior elements with six non-trivial eigenvalues (Subsection 14.2), and we then finally refine the study of pointwise stabilizers in codimension 6 by ruling out SL 2 (F 3 ) (Subsection 14.3).

There finally remain two types of possible junior elements, which are those already appearing in dimension 3 in [START_REF] Oguiso | On the complete classification of Calabi-Yau threefolds of type III 0[END_REF]: diag(1 n-3 , j, j, j) and diag( 4 ).

1 n-3 , ζ 7 , ζ 7 2 , ζ 7
This description of pointwise stabilizers in codimension up to 6 implies that any two junior elements admitting a common fixed point commute. Together with a simple argument about the isogeny type of A (see Section 12), it concludes the proof of Theorem 7.6. In fact, the idea that the existence of certain automorphisms on an abelian variety determines the isomorphism type of some special abelian subvarieties is general ( [START_REF] Shimura | Complex multiplication of abelian varieties and its applications to number theory[END_REF]), and it applies crucially throughout this paper, starting in Chapter 10. From there, it is not so surprising that we are able to determine the isogeny type of A, interpreting the fact that A/G admits a Calabi-Yau resolution as an irreducibility property of the G-equivariant Poincaré decomposition of A.

Under the additional assumption that the group G is abelian, Theorem 7.6 and the results of Section 12 suffice to generalize Theorem 7.5 to higher dimensions, i.e., to the statement that, if A is an abelian variety of dimension n and G is a finite group acting freely in codimension 2 on A such that A/G admits a Calabi-Yau resolution X, then n = 3 and X is X 3 or X 7 .

Also note that G is abelian if and only if any two junior elements g, h of G commute, which by our results can be checked via their matrices acting on a vector space V of dimension 3, 4, 5, or 6. Standard finite group theory allows us to explicitly bound the order of ⟨g, h⟩ depending on this dimension and the isogeny type of A. If the dimension is 3 or 4, the bounds are reasonable enough to launch a computer-assisted search through all possible abstract groups ⟨g, h⟩. Among these, the only groups which, in a faithful 3 or 4-dimensional representation, are generated by two junior elements of the same type, are Z 3 , Z 7 , and the finite simple group SL 3 (F 2 ) of order 168. But a geometric argument on fixed loci excludes SL 3 (F 2 ), whence the wished contradiction. This reproves the classification of [START_REF] Oguiso | On the complete classification of Calabi-Yau threefolds of type III 0[END_REF] in dimension 3, and settles Theorem 7.5.

When V has dimension 5 or 6, we could also bound the order of ⟨g, h⟩ explicitly. For example, we could consider the image of the faithful representation M ⊕ M in SL 2 dim(V ) (Q), and use the classification of irreducible maximal finite integral matrix groups in dimension less than 12 by V. Felsch, G. Nebe, W. Plesken, and B. Souvignier to obtain a bound on the order of ⟨g, h⟩. But the bounds obtained in this way are too large for the SmallGroup library. One needs to better understand the arising matrix groups of larger order, and build a reasonably smaller finite list of possibilities for the abstract group ⟨g, h⟩. It will then remain to figure out geometric ways for ruling out those potential groups in the list other than Z 3 , Z 7 , Z 3 × Z 3 , and Z 7 × Z 7 . Some of our proofs resort to computer-searches among all finite groups of certain fixed orders (relying on the SmallGroup library of GAP). The computer-assisted results used in Subsection 10.3 were actually originally proven by hand using elementary representation theory and Sylow theory. Such arguments being standard in finite group theory, we chose to keep their exposition concise for the sake of readability, and preferred invoking computer-checked facts as black boxes when needed. This approach also has the advantage of better separating abstract group-theoretic arguments on G from properties of the particular representation G → GL(H 0 (T A )). All programs used are available in the Appendix. CHAPTER 8

SOME RESULTS IN MCKAY CORRESPONDENCE

Let us generalize the notion of a junior element, from matrices to automorphisms of abelian varieties. Definition 8.1. If A is an abelian variety of dimension n and g ∈ Aut(A) has finite order, then g can be written as:

g : [z] ∈ A → [M (g)z + T (g)] ∈ A,
where M (g) is a matrix of finite order in GL n (C), T (g) a vector in C n . If g fixes any point a of A, it can be represented locally in a neighborhood of a by its matrix M (g). Hence, it makes sense to say that the automorphism g is junior if g fixes at least one point in A and the matrix M (g) is junior. Remark 8.2. Note that if g ∈ Aut(A) admits a fixed point, then ⟨g⟩ contains no translation, so g and its matrix M (g) have the same order.

Junior elements play a key role in the study of finite quotient singularities, as the previously mentioned Theorem 2.62 emphasizes. Quotient singularities are Qfactorial, so they can not be resolved by small birational morphisms. This yields a simple corollary of Theorem 2.62.

Corollary 8.3. [94]

Let C n /G be a finite Gorenstein quotient singularity, with G acting freely in codimension 1. If the singularity C n /G admits a crepant resolution, then there is a junior element g ∈ G.

In fact, [START_REF] Ito | The McKay correspondence for finite subgroups of SL(3, C). In Higher-dimensional complex varieties[END_REF]Par.4.5] conjectures that under the same hypotheses, if the singularity C n /G admits a crepant resolution, then any maximal cyclic subgroup of G contains a junior element. A counterexample to this conjecture is however presented in Remark 14.15. In this section, we prove a weak version of that conjecture. We phrase it in an analytic set-up, as our later applications call for that, but the proof works in the affine set-up just as well.

Proposition 8.4. Let G ⊂ GL n (C) be a finite group acting freely in codimension 1 on

C n , and let U ⊂ C n be a G-stable simply-connected analytic neighborhood of 0 ∈ C n . If the singularity U/G admits a crepant resolution X G , then the group G is generated by junior elements.

Note that a singularity admitting a crepant resolution is Gorenstein. By [START_REF] Khinich | On the Gorenstein porperty of the ring of invariants of a Gorenstein ring[END_REF] [START_REF] Watanabe | Certain invariant subrings are Gorenstein. I[END_REF], the existence of a crepant resolution X G thus implies that G ⊂ SL n (C).

In order to prove the proposition, we need some background in valuation theory. We refer for that to Section 2.8.

Proof of Proposition 8.4.

Let G be a finite subgroup of GL n (C) acting freely in codimension 1 on C n , and U be a G-stable simply-connected neighborhood of 0 ∈ C n . Suppose that U/G has a crepant resolution X G .

Set G 0 to be the subgroup of G generated by all junior elements. We have the following commutative diagram, constructed from the lower row up:

X X 0 p / / X G q / / U U/G 0 p / / U/G q / / ε ε 0 ε G
The commutative squares containing the normal complex analytic varieties X 0 , X are obtained by taking normalized fibred products. Since quotient singularities are locally Q-factorial, all birational morphisms considered here are divisorial. The morphisms p, q, p, q are finite, and ε, ε 0 , ε G are proper birational.

The key fact is the following.

Lemma 8.5. The prime exceptional divisors of ε 0 are crepant.

Proof. Let E 0 be a prime exceptional divisor of ε 0 , and denote by E its image in X G . Since E is an exceptional divisor of ε G , it is crepant. Let F be a prime exceptional divisor of ε dominating E 0 . By Theorem 2.62 and Remark 2.63, there is a junior element f ∈ G such that v F = v f . Since G 0 is generated by the junior elements of G, we have f ∈ G 0 . We can compute the following ramification index.

|Ram(E 0 /E)| = |Ram(F/E)| |Ram(F/E 0 )| = |Ram(v f , k(U/G)| |Ram(v f ; k(U/G 0 ))| = |⟨f ⟩ ∩ G| |⟨f ⟩ ∩ G 0 | = 1
so E 0 is generically étale over E, hence crepant [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]Prop.5.20].

By this lemma, the finite proper morphism q : X 0 → X G has no ramification divisor. By Zariski purity of the branch locus, since X G is smooth, the morphism q is unramified, hence étale by [81, Ex.III.10.3, Ex.III.10.9].

On the other hand, X G is locally simply-connected by [107, Thm.7.5.2]: There is a contractible neighborhood V of 0 ∈ U/G, such that ε -1 G (V ) is simply-connected. Hence the following commutative diagram.

ε -1 G (V ) q -1 (V ) V q / / ε -1 0 (q -1 (V )) ε 0 ε G q / /
As q is étale, the pre-image ε -1 0 (q -1 (V )) is a disjoint union of deg(q) copies of ε -1 G (V ). Nevertheless, the morphism ε 0 has connected fibers and the base q -1 (V ) is itself connected, hence ε -1 0 (q -1 (V )) is connected, and

deg(q) = |G| |G 0 | = 1,
so G 0 = G and the proof of Proposition 8.4 is settled.

Global result along the same lines.

We close this section with a global result along the same lines as Proposition 8.4.

Lemma 8.6. Let G be a finite group acting freely in codimension 1 on an abelian variety A.

Suppose that A/G has a resolution X G that is simply-connected. Then G is generated by its elements admitting fixed points in A.

Proof. Let G 0 ◁ G be the normal subgroup of G generated by elements admitting fixed points. We want to prove that G 0 = G. We have a commutative diagram:

X 0 ε 0 q / / X G ε G A/G 0 q / / A/G
By definition of G 0 , for every a ∈ A, the stabilizers of a in G and G 0 coincide. Hence, q is étale, and q is étale too by base change. But X G is simply-connected and X 0 is connected, so deg(q) = 1 and G 0 = G.

Remark 8.7. If G is a finite group acting freely in codimension 1 on an abelian variety

A so that A/G has a simply-connected crepant resolution, then G may still contain elements that admit no fixed point. Without loss of generality, we can assume that G contains no translation, up to replacing A by an isogenous abelian variety, but that is the best we can do. CHAPTER 9

THE TWELVE TYPES OF JUNIOR ELEMENTS ON AN ABELIAN VARIETY

Section 8 just shows that, if we want a finite singular quotient of an abelian variety A/G to have a crepant resolution, the group G must contain some junior elements. The fact that in our set-up, G must also act freely in codimension 2 on A is restrictive enough that there are only twelve possibilities for the ranked vector of eigenvalues of a junior element g ∈ G.

Definition 9.1. Let g be a matrix in GL n (C). Assume that it has finite order d.

Since g d = id, g is diagonalizable and has eigenvalues of the form e 2iπa k /d , for integers

a k ∈ [[0, d -1]] satisfying a 1 ∧ . . . ∧ a n ∧ d = 1.
Ordering the integers a k increasingly, we define the ranked vector of eigenvalues of g as the tuple (e 2iπa k /d ) 1≤k≤n .

Proposition 9.2. Let A be an abelian variety of dimension n, and g ∈ Aut(A) be a junior element such that ⟨g⟩ acts freely in codimension 2. Then the order d of g and the ranked vector of eigenvalues of g are in one of the twelve columns of Table 9.1. Table 9.1: Possible ranked vectors of eigenvalues for junior elements in G For d ∈ N, we denote ζ d = e 2iπ/d , and in particular j = e 2iπ/3 and ω = e 2iπ/6 . For k ∈ N, 1 k refers to a sequence of k times the symbol 1 in a row.

d 3 4 6 (e 2iπa k /d ) (1 n-3 , j, j, j) (1 n-4 , i, i, i, i) (1 n-4 , ω, ω, ω, -1) d 6 6 7 (e 2iπa k /d ) (1 n-5 , ω, ω, ω, ω, j) (1 n-6 , ω, ω, ω, ω, ω, ω) 1 n-3 , ζ 7 ,
The proof goes by elementary arithmetic and meticulous case disjunctions. The following terminology should simplify the exposition.

Definition 9.3.

A multiset A is the data of a set A and a function m : A → Z >0 , called the multiplicity function. Intuitively, a multiset is like a set where elements are allowed to appear more than once.

If a multiset A = (A, m) is finite, i.e., its underlying set A = {a 1 , . . . , a k } is finite, we may write A in the following form: {{a 1 , . . . , a 1 m(a 1 ) times , . . . , a k , . . . , a k m(a k ) times }}.

Double-braces are used to avoid confusion between the multiset and the underlying set.

Let A = (A, m) be a finite multiset. If α ∈ Z >0 and , we denote by A * α the multiset (A, αm). If A a subset of Q, and p, q are rational numbers, with q ̸ = 0, we denote by p + qA the multiset (p + qA, m). The cardinal of A is:

|A| := a∈A m(a).
More generally, if f : A → Q is a function, we define: If A = (A, m) and B = (B, n) are two multisets, we define their union: We establish a sequence of three useful lemmas.

A ∪ B := (A ∪ B, 1 A m + 1 B n),
Lemma 9.5. Let u be a positive integer strictly greater than 2. Then we have: [START_REF] Andreatta | Generalized Mukai conjecture for special Fano varieties[END_REF][START_REF] Beauville | Some remarks on Kähler manifolds with c 1 = 0[END_REF]] ∪ {12, 14, 15, 16, 18, 20, 21, 24, 30, 36, 42}.

ϕ(u) 2 u ≤ 8 or Å 2 | u and ϕ(u) 2 u ≤ 4 ã ⇔ u ∈ [[
Proof. Write u = p α 1 1 • p α 2 2 • • • p α k k
, where p 1 < . . . < p k are prime numbers, and α 1 , . . . , α k positive integers, so that:

ϕ(u) 2 u = k i=1 (p i -1) 2 p α i -2 i .
Each of the k factors of this product is greater or equal to 1, unless p α 1 1 = 2 in which case the first factor is 1 2 . Hence, if u satisfies:

ϕ(u) 2 u ≤ 8 or Å 2 | u and ϕ(u) 2 u ≤ 4 ã ,
then each factor satisfies:

(p i -1) 2 p α i -2 i ≤ 8, (9.1)
which yields p i ∈ {2, 3, 5, 7}. Writing u = 2 α 3 β 5 γ 7 δ , where α, β, γ, δ ≥ 0 and using Inequality (9.1) again bounds α ≤ 4, β ≤ 2, γ ≤ 1, δ ≤ 1. Among the finitely many possibilities left, it is easy to check that the solutions exactly are u ∈ [ [START_REF] Andreatta | Generalized Mukai conjecture for special Fano varieties[END_REF][START_REF] Beauville | Some remarks on Kähler manifolds with c 1 = 0[END_REF]] ∪ {12, 14, 15, 16, 18, 20, 21, 24, 30, 36, 42}.

Lemma 9.6. Let u ≥ 2 and d ≥ 3 be integers, such that u divides d. Suppose that there are a positive integer α and a multiset A such that:

A ∪ (d -A) = ßß a ∈ [[1, d -1]] | u = d d ∧ a ™™ * α ,
and such that the quantity:

S A,d (u) := a∈A a u(a ∧ d) satisfies S A,d (u) ≤ 1. Then u, 1 d A, α, S A,d (u) are classified in Table 9.2.
Proof. We consider the following function.

f : a ∈ A ∪ (d -A) → a a ∧ d = ua d .
Clearly, f is an increasing function, and takes values in

{ℓ ∈ [[1, u -1]] | ℓ ∧ u = 1}. It is in fact a bijection, with converse g : ℓ ∈ {ℓ ∈ [[1, u -1]] | ℓ ∧ u = 1} → dℓ u . So |A| ≥ ϕ(u) 2 . The restriction f | A is injective, hence takes at least ϕ(u)
2 distinct values in its image set, so that:

1 ≥ S A,d (u) = 1 u a∈A f (a) ≥ α u Ö 1≤ℓ≤u/2 ℓ∧u=1 ℓ è . (9.2)
Let us denote by Σ(u) the sum 1≤ℓ≤u/2 ℓ∧u=1 ℓ. We have the following coarse estimates:

u ≥ Σ(u) ≥ ϕ(u)/2 ℓ=1 ℓ ≥ ϕ(u) 2 8 , and, if u is even, u ≥ Σ(u) ≥ ϕ(u)/2 ℓ=1 2ℓ -1 ≥ ϕ(u) 2 4 .
Applying Lemma 9.5, these coarse estimates yield finitely many possibilities for u.

Computing explicitly 1 u Σ(u) for the possible values and applying Inequality 9.2 again, we exclude a few of them, finally obtaining that: u ∈ [ [START_REF] Amerik | Morrison-Kawamata cone conjecture for hyperkähler manifolds[END_REF][START_REF] Beauville | Some remarks on Kähler manifolds with c 1 = 0[END_REF]] ∪ {12, 14, 15, 16, 18, 20, 24}.

For each u, we then list by hand the finitely many possibilities for the multiplicity α and the multiset 1 d A, and this is how we construct Table 9.2.

u α

1 d A S A,d (u) ≤ 1 2 1 2 1 2 1 2 , 1 2 1 1 3 , 2 3 1 3 , 2 3 3 1 3 , 1 3 , 1 3 , 2 3 2 3 , 1 1 3 , 1 3 , 1 3 1 1 4 , 1 4
A m ∪ (d m -A m ) = ßß a ∈ [[1, d m -1]] | u m = d m d m ∧ a ™™ * αm .
Suppose additionally that:

k m=1 S Am,dm (u m ) = 1.
Then the data of k and of all u m , α m , 1 dm A m is classified in Table 9.3. Proof. It is easily derived by hand from Table 9.2. S Am,dm (u m ) = 1

u 1 , . . . , u k α 1 , . . . , α k 1 d 1 A 1 , . . . , 1 d k A k freeness in codimension 2 2 1 2 , 1 2 ✗ 2, 3, 6 1, 1, 1 1 2 , 1 3 , 1 6 ✗ 2, 4 1, 1 2 , 1 4 , 1 4 ✗ 2, 6 1, 1 2 , 1 6 , 1 6 , 1 6 ✓ 2, 8 1, 1 2 , 1 8 , 3 8 ✗ 2, 12 1, 1 2 , 1 12 , 5 12 ✗ 3 1 3 , 2 3 ✗ 1 3 , 1 3 , 1 3 ✓ 3, 4, 6 1, 2, 1 1 3 , 1 4 , 1 4 , 1 6 ✗ 3, 6 1, 2 3 , 1 6 , 1 6 ✗ 1, 1 3 , 1 6 , 1 6 , 1 6 , 1 6 ✓ 2, 1 3 , 1 3 , 1 6 , 1 6 ✗ 3, 12 1, 1 3 , 1 12 , 7 12 ✗ 4 1 4 , 3 4 ✗ 1 4 , 1 4 , 1 4 , 1 4 ✓ 4, 6 2, 1 4 , 1 4 , 1 6 , 1 6 , 1 6 ✗ 4
We can now prove Proposition 9.2.

Proof of Proposition 9.2. Denote by d the order of the junior element g, by (e 2iπa j /d ) 1≤j≤n its ranked vector of eigenvalues, and by P (g) the characteristic polynomial of its matrix M (g). As g itself acts freely in codimension 2 and g is junior, it must be that d ≥ 3. By Lemma 2.76, there are positive integers k, (u m ) 1≤m≤k ordered increasingly, and (α m ) 1≤m≤k , such that:

n j=1 (X -e 2iπa j /d )(X -e 2iπa j /d ) = P (g)P (g) = k m=1 Φ um αm . (9.3)
Note that Φ um (e 2iπa j /d ) = 0, or equivalently Φ um (e 2iπa j /d ) = 0, if and only if u m = d d∧a j . We define the following partition of [ [1, n]]

for m ∈ [[1, k]], I m := {j ∈ [[1, n]] | u m = d d∧a j }; A m := {{a j | j ∈ I m }}, as a multiset.

By Identity 9.3, for m ∈ [[1, k]] we have:

A m ∪ (d -A m ) = {{r ∈ [[1, d -1]] | u m = d d ∧ r = 0}} * αm (9.4)
Moreover, since g is junior:

1 = n j=1 a j d = k m=1 j∈Im a j d = k m=1 j∈Im a j u m (d ∧ a j ) = k m=1
S Am,d (u m ). (9.5) So, possibly leaving out the data of index 1, if u 1 = 1 (which is determined by the multiplicity α 1 ∈ N, since then A 1 = {{0 α 1 }} and S A 1 ,d (u 1 ) = 0), Lemma 9.7 applies, showing that there are finitely many possibilities for

k, (u m ) 1≤m≤k , (α m ) 1≤m≤k , Å 1 d A m ã 1≤m≤k
and listing them. We exclude by hand a lot of these possibilities using the assumption that ⟨g⟩ acts freely in codimension 2 on A, i.e., that for all ℓ ∈ [[1, d -1]], there must be distinct indices j 1 (ℓ), j 2 (ℓ),

j 3 (ℓ) ∈ [[1, n]],
such that none of the

ℓa j i (ℓ) d
is an integer. What remains then is precisely the list in Table 9.1.

CHAPTER 10

CYCLICITY OF THE POINTWISE STABILIZERS OF LOCI OF CODIMENSION 3 AND 4

We now know that G is generated by junior elements, which we have classified into twelve different types. However, this is by far insufficient to determine the structure of G. Even locally, for W ⊂ A a subvariety, the pointwise stabilizer PStab(W ) := {g ∈ G | ∀w ∈ W, g(w) = w} could as well be cyclic and generated by one junior element, as it could be more complicated, e.g., if it contained non-commuting junior elements.

In this section, we show that in fact, if W has codimension 3 or 4 in A, PStab(W ) is trivial or cyclic, generated by one junior element. Let us outline the proof. Subsection 10.1 reduces to proving this in the case when W is a point in an abelian variety B of dimension 3 or 4. Up to conjugating the whole group G by a translation, we therefore just work on the case W = {0}. Assuming PStab(W ) is not trivial, we can then find a junior element g ∈ PStab(W ), that is of one of the twelve types of Section 9. Subsection 10.2 exhibits a correlation between the type of g and the isogeny type (possibly even isomorphism type) of the abelian variety B on which it acts. A corollary is that if g, h ∈ PStab(W ) are two junior elements, then they should either have the same type, or one is of type (1 n-4 , ω, ω, ω, -1) and the other (1 n-3 , j, j, j), or one is of type (1 n-4 , i, i, i, i) and the other (1 n-4 , ζ 12 , ζ 12 , ζ 5 12 , ζ 5 12 ). In particular, if PStab(W ) is cyclic, it must indeed be generated by one junior element. The conclusive Subsection 10.3 is the most technical. For any given abelian three-or fourfold B of one of the types just defined, we classify all finite subgroups of Aut(B, 0) := {f ∈ Aut(B) | f (0) = 0, i.e., T (f ) = 0} that act freely in codimension 2 on B and are generated by junior elements. The main idea is to bound the order of such groups, to scrutinize the finite list arising, and to rule out all but the cyclic case of the list by the assumption on generators.

Reduction to a 3 or 4-dimensional question.

Definition 10.1. Let A be an abelian variety. An abelian subvariety of A is a closed subvariety of A that is also a subgroup of the abelian group (A, +). A translated abelian subvariety of A is the image by a translation of an abelian subvariety of A.

We say that two translated abelian subvarieties B and C of A are complementary if one of the following equivalent statements hold: (i) B ∩ C is non-empty and, for some p ∈ B ∩ C, it holds:

H 0 ({p}, T B ) ⊕ H 0 ({p}, T C ) = H 0 ({p}, T A ).
(ii) The addition map i : B × C → A is an isogeny.

Proof. (i) ⇒ (ii): as the translation by (p, p), respectively by 2p, is an isomorphism from B × C to (B -p) × (C -p), respectively of A, it is enough to prove the statement for p = 0. As dim(A) = dim(B × C) and the varieties are regular, we simply check that i is quasi-finite. Since B ∩ C is the intersection of two abelian subvarieties of A satisfying:

H 0 ({0}, T B ) ∩ H 0 ({0}, T C ) = {0}, the set B ∩ -C is discrete in A, hence finite. For a ∈ Im(i), say a = i(a B , a C ), we can express the fiber i -1 (a) = {(b + a B , -b + a C ) | b ∈ B ∩ -C},
so it is finite, and i is indeed quasi-finite.

(ii) ⇒ (i): fix c 0 ∈ C. The addition i is onto, so let (p, c) ∈ B × C be such that p + c = 2c 0 . Clearly, p = 2c 0 -c ∈ B ∩ C, and as i is locally analytically an isomorphism,

H 0 ({p}, T B ) ⊕ H 0 ({p}, T C ) = H 0 ({2p}, T A ) = H 0 ({p}, T A ).
Remark 10.2. If B and C are complementary translated abelian subvarieties of an abelian variety A, and t ∈ A is any point, then B + t and C are complementary as well. Our notion of complementarity is weaker than the notion defined for abelian subvarieties in [17, p.125].

Let us now state our reduction result. Note that it applies not only in codimension 3 and 4, but in any higher codimension as well.

Proposition 10.3. Let A be an abelian variety, G be a finite group acting freely in codimension 2 on A. Suppose that the quotient A/G admits a crepant resolution. Let W be a subvariety of codimension m in A such that PStab(W ) ̸ = {1}. Then:

(1) For any t ∈ W there is a translated abelian subvariety B of A which is PStab(W )stable, contains t, and is complementary to W in A.

( We now prove (2): let g ∈ PStab(W ). As g fixes all points of B ∩ W , its restriction g| B has a fixed point. As g(B) = B, we have:

) 2
M (g) = Å id dim(W ) 0 0 M (g| B ) ã ,
and thus g is indeed junior if and only if g| B is. We move on to (3). Take a general point w ∈ W such that PStab(w) = PStab(W ). Since PStab(w) is finite, any analytic neighborhood of w in A contains a contractible analytic neighborhood U of w that is PStab(w)-stable. Up to reducing it even more, we can assume that for any g ∈ G \ PStab(w), g(U ) ∩ U = ∅. So, an analytic neighborhood of [w] ∈ A/G is biholomorphic to U/PStab(w). Hence, Proposition 8.4 applies and PStab(w) is generated by junior elements.

The abelian varieties corresponding to the twelve juniors.

Let A be an abelian variety of dimension n, G be a finite group acting freely in codimension 2 on A such that A/G has a crepant resolution. By Corollary 8.3, G ⊂ Aut(A) must entail a junior element presented in Table 9.1 (up to its translation part, and up to similarity for its linear part). The fact that, in some coordinates, a given matrix of Table 9.1 acts as an automorphism on the abelian variety A imposes some restrictions. Using the theory of abelian varieties with complex multiplication, these restrictions are investigated by Proposition 10.6. Notation 10.4. Let us defined the following quadratic integers

u 7 = -1 + i √ 7 2 , u 8 = i √ 2, u 15 = 1 + i √ 15 2 , u 20 = i √ 5, u 24 = i √ 6,
and the following algebraic integers, whose square are quadratic integers

u 16 = i » 4 + 2 √ 2, v 16 = i » 4 -2 √ 2.
For z ∈ C \ R, we define the elliptic curve With these notations, we can state the main result of the subsection.

Proposition 10.6. Let A be an abelian variety. Suppose that there is a junior element g ∈ Aut(A), and that ⟨g⟩ acts freely in codimension 2 on A. Denote by W an irreducible component of Fix(g) := {a ∈ A | g(a) = a}. Let B be a complementary to W in A. Then the isogeny type of B is entirely determined by the type of the junior element g by Table 10.1, unless g is of type (1 n-4 , ω, ω, ω, -1). Moreover, the isomorphism type of a ⟨g⟩-stable complementary B st to W in A is also entirely determined by the type of g, unless g is of type (1 n-4 , ω, ω, ω, -1) or (1 n-5 , ω, ω, ω, ω, j).

Notation 10.7. Let V be a C-vector space, f : V → V be a linear map. We denote by EVal(f ) the set of eigenvalues of f in C, by EVal(f ) the multiset of eigenvalues of f in C counted with multiplicities. If λ ∈ EVal(f ), we denote by E f (λ) the eigenspace of f for the eigenvalue λ. We denote by Z(Φ d ) ⊂ U d the set of primitive d-th roots of unity in C.

Let us first carry out an important computation, that makes plain where these special types of abelian varieties come from. Let k ≥ 3 be an integer. There is a natural action of

ζ k ⊗ 1 on the algebra Z[ζ k ] ⊗ C. We compute its eigenvalues. type of g isogeny type of B isomorphism type of B st (1 n-3 , j, j, j) E j 3 E j 3 (1 n-4 , i, i, i, i) E i 4 E i 4 (1 n-4 , ω, ω, ω, -1) E × E j 3 ,
Z[ζ k ] ⊗ C = ξ∈Z(Φ k ) E ζ k ⊗1 (ξ) (10.1) Now, consider a subset S k of Z(Φ k ) such that S k S k = Z(Φ k ).
For example, if we let g be a junior element of one of the twelve types in Table 9.1, and we assume that g has an eigenvalue of order k, we could set

S k = S k (g) = EVal(g) ∩ Z(Φ k ). This defines a Z-linear inclusion f (S k ) : Z[ζ k ] → ξ∈S k E ζ k ⊗1 (ξ) ≃ C ϕ(k)/2 (10.2) It is worth noting that the Z-linear inclusion f (S k ) ⊕ f (S k ) corresponds to the natural inclusion of Z[ζ k ] in Z[ζ k ] ⊗ C given by Identity 10.1.
The following lemma is key.

Lemma 10.8. If S k = S k (g) for a junior element g of Table 9.1, then the corresponding abelian variety Proof of Lemma 10.8.

C ϕ(k)/2 /Im(f (S k )) is described in
Let F = Q[ζ k ], r = ϕ(k) 2 .
Let us define {φ i } 1≤i≤r : composing f (S k ) defined in Identity 10.2 with the projections on the r eigenspaces, we obtain morphisms of Z-algebras f i : Z[ζ k ] → C, which we tensor by Q and normalize to define morphisms of Q-algebras:

φ i = 1 f i (1) (f i ⊗ 1) : Q[ζ k ] → C.
By Identities 10.1 and 10.2, the morphisms {φ i , φ i } 1≤i≤r are linearly independent over Q, whereas the morphisms {φ i } 1≤i≤r define an embedding of F into the Q-algebra of linear endomorphisms of the abelian variety C ϕ(k)/2 /Im(f (S k )). In other words, the abelian variety C ϕ(k)/2 /Im(f (S k )) has CM-type (F, {φ i } 1≤i≤r ). This is in fact the sole abelian variety with this CM-type, by [START_REF] Masley | Cyclotomic fields with unique factorization[END_REF], [START_REF] Shimura | Complex multiplication of abelian varieties and its applications to number theory[END_REF]Prop.17] remembering that k ∈ {7, 15, 16, 20, 24}.

Applying Lemma 2.97 with

K = Q[u k ],
we get the wished description of the abelian variety C ϕ(k)/2 /Im(f (S k )), by an easy verification involving that:

• u 7 = ζ 7 + ζ 2 7 + ζ 4 7 ,
• The next result follows almost effortlessly from the ideas of [155, p.333-334].

Lemma 10.10. Let B be an abelian variety. Suppose that there is an automorphism g of B whose set of eigenvalues is one of the S k in Table 10.2. Then B is isomorphic to a power of the abelian variety C ϕ(k)/2 /Im(f (S k )).

Proof. Let Λ be a lattice in C n such that B = C n /Λ. The linear action of g restricting to Λ, it provides it with a Z[g]-module, i.e., a Z[ζ k ]-module structure, since the minimal polynomial of g is Φ k . As such, Λ is finitely-generated and torsion-free. But by [START_REF] Masley | Cyclotomic fields with unique factorization[END_REF], since k ∈ [ [START_REF] Andreatta | Generalized Mukai conjecture for special Fano varieties[END_REF][START_REF] Borcea | Homogeneous vector bundles and families of Calabi-Yau threefolds. II. In Several Complex Variables and Complex Geometry[END_REF]] ∪ {24}, the ring of cyclotomic integers Z[ζ k ] is a principal ideal domain. So, by the structure theorem for finitely-generated modules over principal ideal domains, Λ ≃ Z[ζ k ] 2n/ϕ(k) , and the action of g on Λ identifies with the multiplication by

ζ k on Z[ζ k ] 2n/ϕ(k) .
The embedding Λ → H 0 (B, T B ) ≃ C n can be recovered from the action of g on Λ. Indeed, there is an induced action of g ⊕ g on Λ ⊗ C = H 0 (B, T B,R ⊗ C) ≃ C 2n . This action splits into two blocks: g is acting on H 0 (B, T B ) and g is acting on its supplementary conjugate in H 0 (B, T B,R ⊗ C). By the requirement on its set of eigenvalues S k , g has no eigenvalue in common with g, and therefore:

H 0 (B, T B ) = ξ∈EVal(g) E g⊕g (ξ).
Hence, the corresponding embedding Z[ζ k ] 2n/ϕ(k) → C n must similarly be given by:

C n = ξ∈EVal(g) E ζ k ⊗1 (ξ),
where ζ k ⊗1 is the action by componentwise multiplication on Z[ζ k ] 2n/ϕ(k) ⊗C. In other words, this embedding is the blockwise embedding f (S k ), repeated on 2n ϕ(k) blocks of dimension ϕ(k)

2 each. So B ≃ C ϕ(k)/2 /Im(f (S k )) 2n/ϕ(k) .
The proof of Proposition 10.6 is now easy.

Proof of Proposition 10. , ω, ω, ω, -1) or (1 n-5 , ω, ω, ω, ω, j), then g| Bst has eigenvalues of two different orders. By [START_REF] Birkenhake | Complex abelian varieties[END_REF]Thm.13.2.8], there are then two ⟨g⟩stable complementary translated abelian subvarieties B 1 and B 2 in B st , such that all eigenvalues of g| B 1 have order k 1 = 6, and all eigenvalues of g| B 2 have the same order k 2 < 6. By definition, B st is isogenous to B 1 × B 2 , and thus its isogeny type can be derived from the isomorphism types of B 1 and B 2 , given by Lemma 10.10 if k 1 , k 2 ≥ 3. However, if g is of type (1 n-4 , ω, ω, ω, -1), then k 2 = 2 and B 2 can be any elliptic curve, and that is why the isogeny type of B st is not entirely determined in this case.

On the other hand, if g is of any other type, then all eigenvalues of g| Bst are of the same order k ≥ 3, and Lemma 10.10 determines the isomorphism type of B st .

Group theoretical treatment of a point's stabilizer in dimension 3 or 4.

We can now establish the following proposition.

Proposition 10.11. Let A be an abelian variety, G ⊂ Aut(A) be a finite group acting freely in codimension 2. Suppose that the quotient A/G admits a crepant resolution.

Let W be a subvariety of codimension m ≤ 4 in A such that PStab(W ) ̸ = {1}. Then PStab(W ) is a cyclic group generated by one junior element. By Proposition 10.3, it reduces to proving the following result. Proposition 10.12. Let B be an abelian variety of dimension m ≤ 4, F ⊂ Aut(B, 0) be a finite group acting freely in codimension 2 and fixing 0 ∈ B. Suppose that F is generated by junior elements. Then F is a cyclic group generated by one junior element.

We refer the reader to [START_REF] Robinson | A course in the theory of groups[END_REF], [START_REF] Isaacs | Finite group theory[END_REF] for standard facts in finite group theory, and in particular Sylow theory and representation theory. Let us just recall a few notations used in the following. Notation 10.13. We denote by C F (H), respectively N F (H), the centralizer, respectively normalizer, of a subset H of a group F , i.e.,

C F (H) := {f ∈ F | ∀ h ∈ H, f h = hf } N F (H) := {f ∈ F | f H = Hf }
If H has a single element or is a subgroup of F , then C F (H) and N F (H) are subgroups of F . Notation 10.14. Let F be a finite group, V be a vector space of finite dimension, ρ : F → GL(V ) be a group morphism, i.e., a faithful representation of F in V . The character χ of ρ is the map χ : f ∈ F → Tr(ρ(f )) ∈ C * . By Schur's lemma, the representation ρ decomposes as a direct sum of irreducible representations:

ρ = ρ ⊕n 1 1 ⊕ . . . ⊕ ρ ⊕n k k ,
and, accordingly, if χ i denotes the character of ρ i , we have χ = n 1 χ 1 + . . . + n k χ k . By orthogonality of the irreducible characters,

⟨χ, χ⟩ = (n 2 1 + . . . + n 2 k )|F |.
We refer to u = n 2 1 + . . . + n 2 k as the splitting coefficient of the representation ρ.

We start proving lemmas towards Proposition 10.12. The first lemma classifies all possible finite order elements in Aut(B, 0) of determinant one acting freely in codimension 2, when B is an abelian fourfold. Lemma 10.15. Let B be an abelian fourfold, and g ∈ Aut(B, 0) be a finite order element such that ⟨g⟩ acts freely in codimension 2 on B. Then the order of g and the matrix of a generator of ⟨g⟩ are given in Table 10.3, together with the restrictions on B, if any.

Proof. Let ζ be an eigenvalue of g of order u, such that (ϕ(u), u) is maximal in N 2 for the lexicographic order. By Lemma 2.76, Φ u divides the characteristic polynomial χ g⊕g in Q[X], so ϕ(u) ≤ 2 dim B = 8. Let us discuss cases:

(1) If ϕ(u) = 1, then u = 1 or 2. As g acts freely in codimension 2 and has determinant one, g = ±id B .

(2) Suppose that ϕ(u) = 8. Then g has four distinct eigenvalues of order u, and hence has order u. Listing integers of Euler number 8, u ∈ {15, 16, 20, 24, 30}.

There is a generator g ′ of ⟨g⟩ of which e 2iπ/u is an eigenvalue. Denote its other eigenvalues by e 2iπa/u , e 2iπb/u , e 2iπc/u , with

• a, b, c ∈ [[1, u -1]] coprime to u • u divides 1 + a + b + c • and Φ u (X) =(X -e 2iπ/u )(X -e 2iπ(u-1)/u )(X -e 2iπa/u )(X -e 2iπ(u-a)/u ) (X -e 2iπb/u )(X -e 2iπ(u-b)/u )(X -e 2iπc/u )(X -e 2iπ(u-c)/u )
We check by hand the solutions to this system and plug them in 3) Suppose that ϕ(u) = 6. Then g has three distinct eigenvalues of order u, and one eigenvalue of order v, with ϕ(v) = 1 or 2. Since g u has three trivial eigenvalues and ⟨g⟩ acts freely in codimension 2, g u = id B , so g has order u and v divides u. Listing the integers of Euler number 6, u ∈ {7, 9, 14, 18}. Using that ).

χ g⊕g = Φ u Φ v or Φ u Φ v 2 ,
(4) Suppose that ϕ(u) = 4. Then g has two distinct eigenvalues of order u, and two remaining eigenvalues of respective order v 1 ≤ v 2 . As ⟨g⟩ acts freely in codimension 2, g u , which has two trivial eigenvalues, must be trivial, so g has order u and v 1 and v 2 divide u. Similarly, g lcm(v 1 ,v 2 ) = id B , so u divides lcm(v Listing possibilities by hand, we see that (v 1 , v 2 ) ∈ {(2, 3), [START_REF] Andreatta | Generalized Mukai conjecture for special Fano varieties[END_REF][START_REF] Andreatta | Special rays in the Mori cone of a projective variety[END_REF], (4, 6)}.

From the divisibility relations between v 1 , v 2 and u, we obtain that u = 12, and in fact, (v 1 , v 2 ) = [START_REF] Andreatta | Generalized Mukai conjecture for special Fano varieties[END_REF][START_REF] Andreatta | Special rays in the Mori cone of a projective variety[END_REF] or [START_REF] Andreatta | Special rays in the Mori cone of a projective variety[END_REF][START_REF] Auffarth | Smooth quotients of complex tori by finite groups[END_REF]. In particular, g has order 12, so g 6 = -id B , and so g 3 has four eigenvalues of order 4. But since v 1 = 3 or v 2 = 6, this can not be the case. Contradiction!

(5) The last case is when ϕ(u) = 2, i.e., u = 3, 4, or 6. In that case, each eigenvalue of g has order 1, 2, 3, 4, or 6. As ⟨g⟩ acts freely in codimension 2, g has at most one eigenvalue of order 1 or 2.

(a) Suppose that g has an eigenvalue of order 4. As it has determinant 1, it has an even number of eigenvalues of order 4, so at least two of them. Hence, by freeness in codimension 2, g 4 = id B , and so g 2 = -id B , i.e., all eigenvalues of g have order 4. There is a generator of ⟨g⟩ similar to either diag(i, i, i, i), or diag(i, -i, i, -i).

(b) Suppose that u = 3. Then as (ϕ(v), v) ≤ (ϕ(u), u) for any order v of another eigenvalue of g, the other eigenvalues have order 1, 2, or 3. Hence, there are at least three eigenvalues of order 3, and thus by freeness in codimension 2, g 3 = id B . So g has order 3 and there is a generator of ⟨g⟩ similar to either diag(1, j, j, j), or diag(j, j 2 , j, j 2 ).

(c) Suppose finally that u = 6 and g has no eigenvalue of order 4: Then g has order 6, so g 3 = -id B . All eigenvalues of g thus have order 2 or 6, so g has at least three eigenvalues of order 6. As g has determinant 1, we only have two possibilities: There is a generator of ⟨g⟩ similar to diag(-1, ω, ω, ω), or diag(ω, ω 5 , ω, ω 5 ).

This discussion constructs the first two columns of the table. The restrictions on B given in the third column are given by the same arguments as in the proof of Lemmas 10.8, 10.10.

Corollary 10.16. Let B be an abelian fourfold, and let g, h ∈ Aut(B, 0) be junior elements such that ⟨g⟩ and ⟨h⟩ act freely in codimension 2, with ord(g) ≤ ord(h). Then there are three possibilities:

• g and h are similar, in particular have the same order;

• g is similar to diag(1, j, j, j), h is similar to diag(-1, ω, ω, ω), and B is isogenous to E × E j 3 for some elliptic curve E; ). So g and h are similar. The same argument works if g has order 8, 15, 16, 20, 24. If g has order 3 or 6, then by Lemma 10.15, B is isogenous to E × E j 3 for some elliptic curve E. By uniqueness in the Poincaré decomposition of B [START_REF] Birkenhake | Complex abelian varieties[END_REF]Thm.5.3.7], B is not isogenous to any of the other special abelian varieties appearing in Lemma 10.15. So, by Lemma 10.15 again, h being junior must have order 3 or 6. As we assumed ord(g) ≤ ord(h), the only strict inequality is when g has order 3 and h has order 6. In this case, by Proposition 9.2, g is similar to diag(1, j, j, j) and h to diag(-1, ω, ω, ω).

• g = iid B ,
The same argument works if g has order 4 or 12.

We can now prove cyclicity of F when it contains a junior element of order 3.

Proposition 10.17. Let B be an abelian fourfold, and let F be a finite subgroup of Aut(B, 0) acting freely in codimension 2, generated by junior elements. Suppose that F contains an element similar to diag(1, j, j, j). Then F is cyclic and generated by one junior element.

Proof. By Corollary 10.16, B is isogenous to E × E j 3 for some elliptic curve E, and any junior element in Aut(B, 0) is similar to diag(1, j, j, j), or diag(-1, ω, ω, ω).

Suppose by contradiction that F is not generated by one junior element. Then there are two junior elements g, h ∈ F such that ⟨g⟩ ⊈ ⟨h⟩ and ⟨h⟩ ⊈ ⟨g⟩. Up to possibly replacing them by their square, we have g and h both similar to diag(1, j, j, j). Their eigenspaces satisfy dim E g(j) ∩ E h(j) = 2 ≤ dim E gh-1 (1). As ⟨g, h⟩ ⊂ F acts freely in codimension 2, g = h. Since ⟨g⟩ ⊂ ⟨h⟩, g ̸ = g, so g = g 2 . Similarly, h = h 2 . Since g 3 = h 3 = -id, it nonetheless yields g = h, contradiction.

Let us now present our general strategy to prove that F is cyclic. By Lemma 10.15, the prime divisors of |F | are 2, 3, 5, and 7. Hence, |F | = 2 α • 3 β • 5 γ • 7 δ . Since 2 α (respectively 3 β , etc.) is the order of a 2 (respectively 3, etc.)-Sylow subgroup of F , we can rely on Sylow theory to bound |F |, as in the following result. Proof. As S is a p-group, its center Z(S) is non-trivial. Hence, it contains an element g of order p. Let h ̸ = id ∈ S. By Lemma 10.15, F has no element of order p 2 , so h has order p. Since g and h commute, they are codiagonalizable. Let v, w be two non-colinear common eigenvectors of them associated with eigenvalues other than 1. Let g ∈ ⟨g⟩ and h ∈ ⟨h⟩ satisfy g

(v) = h(v) = ζ p v.
If p = 3 or 5, Lemma 10. [START_REF] Beltrametti | A view on extending morphisms from ample divisors[END_REF] shows that E g (1) = E h (1) = {0}, so gh -1 can not have 1 as an eigenvalue and be of order p. So it is trivial, i.e., g = h, and h ∈ ⟨g⟩.

Suppose p = 7. If g(w) ̸ = h(w), then by Lemma 10. Proof. By Lemma 10.15, the element of order 2 in F is unique: it is -id B . By [170, 5.3.6], S is hence either cyclic or a generalized quaternion group. Moreover, by Lemma 10.15, S has no element of order 32. Hence, the only case where the order of S does not divide 16, is when S is isomorphic to Q 32 . Let us however show that this is impossible.

Indeed, Q 32 contains an element h of order 16 and an element s of order 4 such that shs -1 = h -1 [170, pp.140-141]. However, if h ∈ S is an element of order 16, it can not be conjugated in S to h -1 , because by Lemma 10.15 they have distinct eigenvalues.

Proof of Proposition 10.18. It is straightforward from Lemma 10.19 and Lemma 10.20.

The following Lemma and Proposition show that if 7 divides |F |, i.e., if F contains a junior element of order 7, then F is cyclic generated by one junior element of order 7.

Lemma 10.21. Let B be an abelian fourfold, and let F be a finite subgroup of Aut(B, 0) acting freely in codimension 2, containing no junior element of order 3. Suppose that 7 divide |F |. Let S be a 7-Sylow subgroup of F . Then there is a normal subgroup N of F such that F = N ⋊ S.

Proof. By Burnside's normal complement theorem [START_REF] Rotman | An introduction to the theory of groups[END_REF]Theorem 7.50], it is enough to show that N F (S) = C F (S).

Let g be a generator of S. By Lemma 10.15, if f ∈ N F (S), then f gf -1 ∈ {g, g 2 , g 4 }, because they are the only elements with the same set of eigenvalues as g. So f 3 ∈ C F (S). Let us show by contradiction that f ∈ N F (S) can not have order 3. Looking at the action of f on the eigenspaces of g in coordinates diagonalizing g, f = Ü t 0 0 0 0 0 z 0 0 0 0 y 0 x 0 0 ê , with xyz = t, and so χ f = (X -t)(X 3 -t). But by Lemma 10.15, elements of order 3 in F (which by assumption cannot be junior) have characteristic polynomial (X 2 + X + 1) 2 , contradiction. So N F (S) has no element of order 3. To sum up, if f ∈ N F (S), then f 3 ∈ C F (S) and 3 is prime to the order of f , so f ∈ C F (S). Else, the junior elements of F each have order 4 or 12. If there are no junior elements of order 12, Lemma 10.23 concludes again. If there is a junior element g of order 12, then a quick computation from Lemma 10.15 shows that g 3 is the only junior element of order 4 in F , and thus the junior elements of order 12 actually generate F too, so we conclude by Lemma 10.23. These versions of Lemma 10.21 for 3-and 5-Sylow subgroups will be useful too. We finally prove the following two key propositions, which imply Proposition 10.12.

F ≃ (Z 5 γ ⋊ Z 3 β ) ⋊ Z 2 α . Since 3 β
is coprime to ϕ(5 γ ), the group Z 5 γ has no automorphism of order 3, and thus the first semidirect product is direct:

F ≃ (Z 5 γ × Z 3 β ) ⋊ Z 2 α .
no element of order 4 in their centers, so g has order 8. As Q 8 has no element of order 8, S is isomorphic to Q 16 . But we easily check with GAP that:

• The irreducible representations of Q 16 have rank 1 or 2.

• The only irreducible representations of Q 16 of rank r sending the unique element of order 2 to -id r are two complex conjugates representations with r = 2, so all elements of S ⊂ Aut(B, 0

) have characteristic polynomials in Q[X].
However, g ∈ S is a junior element of order 8, which by Lemma 10.15 has a non-rational characteristic polynomial, contradiction. So any junior element in F has order 15, 20 or 24. We also know that:

• F has exactly one element of order 2, by Lemma 10.15. We check with GAP that there are only five groups satisfying all these properties, namely the groups indexed [START_REF] Cynk | Generalised Kummer constructions and Weil restrictions[END_REF][START_REF] Andreatta | Special rays in the Mori cone of a projective variety[END_REF], [START_REF] Cynk | Generalised Kummer constructions and Weil restrictions[END_REF][START_REF] Beauville | Variétés kählériennes dont la première classe de Chern est nulle[END_REF], [START_REF] Hartshorne | Ample subvarietes of algebraic varieties[END_REF][START_REF] Bonavero | On covering and quasicovering families of curves[END_REF], [START_REF] Di Cerbo | Birational boundedness of low-dimensional elliptic Calabi-Yau varieties with a section[END_REF][START_REF] Bakker | Algebraic approximation and the decomposition theorem for Kähler Calabi-Yau varieties[END_REF], and [START_REF] Di Cerbo | Birational boundedness of low-dimensional elliptic Calabi-Yau varieties with a section[END_REF][START_REF] Campana | Connexité rationnelle des variétés de Fano[END_REF] in the SmallGroup library. The function StructureDescription then shows that they are respectively of the form

• A 2-Sylow subgroup of F is isomorphic to Q 8 or Q 16 ,
Z 5 ⋊ Q 8 , Z 5 × Q 8 , Z 5 ⋊ Q 16 , Z 3 ⋊ Q 16 , and Z 3 × Q 16 . Note that only Z 5 × Q 8 , Z 5 ⋊ Q 16
are generated indeed by their elements of orders (15, 24, or) 20. Checking the irreducible character tables of these two cadidates with GAP shows that they have no appropriate four-dimensional representation (see Appendix for programs supporting this discussion.) This concludes the proof of Proposition 10.27.

Proof of Proposition 10.12. If F contains a junior element of order 3, then Proposition 10.17 applies and shows that F is cyclic generated by one junior element. If F contains no junior element of order 3, but one of order 7, then Proposition 10.22 applies and shows that F is cyclic generated by one junior element. Finally, if F contains no junior element of order 3 or 7, Proposition 10.27 shows that its 2-Sylow subgroups are cyclic or trivial, Proposition 10.28 deduces that F is cyclic and Corollary 10.24 proves that F is generated by one junior element.

order of g a generator of ⟨g⟩ up to similarity restrictions on B id -id diag(j, In light of this remark, the proof of Proposition 11.1 must crucially involve global arguments.

j 2 , j, j 2 ) diag(i, -i, i, -i) diag(ζ 5 , ζ 2 5 , ζ 3 5 , ζ 4 5 ) B arbitrary diag(ω, ω 5 , ω, ω 5 ) diag(ζ 8 , ζ
) diag(1, j, j, j) B ∼ E × E j 3 diag(-1, ω, ω, ω) diag(j 2 , ζ 9 , ζ 4 9 , ζ 7 9 ) B ≃ E j 4 18 diag(ω 5 ,
11.1 Ruling our junior elements of order 4,8,12,15,16,20,24. In this subsection, we rule out the seven types of junior elements or order other than 3, 6, 7.

Proposition 11.3. Let A be an abelian variety, G a group acting freely in codimension

2 on A such that A/G has a crepant resolution X. Then any junior element of G has order 3, 6, or 7.

Remark 11.4. Let A be an abelian variety, G be a group acting freely in codimension 2 on A. As translations in G form a normal subgroup G 0 , we can write:

(A/G 0 )/(G/G 0 ) ≃ A/G.
Clearly, A/G 0 is isogenous to A and G/G 0 still acts freely in codimension 2 on it, except that it contains no translation. Hence, we can assume without loss of generality that G contains no translation (other than id). In particular, any element of G has the same finite order as its matrix.

Proof of Proposition 11.3. By contradiction, suppose that g ∈ G is a junior element of order d ∈ {4, 8, 12, 15, 16, 20, 24}, of minimal order among the junior elements of G of such orders. Up to conjugating the whole group G by an appropriate translation, we may assume that g fixes 0 ∈ A. In particular, g fixes pointwise an abelian subvariety W of A of codimension 4, so Propositions 10.11 and 10.3 show that PStab(W ) = ⟨g⟩, and define a ⟨g⟩-stable complementary abelian subvariety B to W in A. The key to the proof is that a well-chosen power g α of g has strictly more fixed points in B than g, as many distinct eigenvalues as g, but is not be a junior element. Indeed, we set α depending on d as follows, and check with Proposition 9.2 that g α is not junior and has as many distinct eigenvalues as g. As for fixed points, applying [START_REF] Birkenhake | Complex abelian varieties[END_REF]Prop.13 Let τ ∈ B be a fixed point of g α that is not fixed by g. Note that W +τ is pointwise fixed by g α . By Proposition 10.11, PStab(W + τ ) = ⟨h⟩ for some junior element h. By Proposition 10.3, there is an ⟨h⟩-stable translated abelian subvariety B ′ of A containing τ such that B ′ and W + τ are complementary. By uniqueness in Poincaré's complete reducibility theorem [START_REF] Birkenhake | Complex abelian varieties[END_REF]Thm.5.3.7], the abelian varieties B and B ′ -τ are isogenous, hence determined by the order of g and h respectively, by Lemma 10.15.

Let us discuss the special case when B ≃ E i 4 , i.e., when junior elements of order both 4 and 12 exist in Aut Q (B, 0) = Aut Q (B ′ , 0). If g or h has order 4, then by the minimality assumption on g, g has order 4, and by Lemma 10.15, either g = h or g 3 = h. So g ∈ ⟨h⟩, and thus g(τ ) = τ , contradiction! By Corollary 10.16, we can now assume that g and h have the same order d ∈ {8, 12, 15, 16, 20, 24}, and similar matrices. Recall that g α ∈ ⟨h⟩. Since g and h have the same order, it implies ⟨g α ⟩ = ⟨h α ⟩, i.e., g α = h uα for some u coprime to d α . Since g and g α , and h and h uα have the same number of distinct eigenvalues, it follows from g α = h uα that the eigenspaces of g and h are the same, i.e., g and h commute. We discuss two cases separately.

(1) If d = 8 or 12, then in appropriate coordinates, we have:

M (g) = diag(1 n-4 , ζ d , ζ d , ζ d m , ζ d m ) M (h) = diag(1 n-4 , ζ d m , ζ d m , ζ d , ζ d ) for some integer m ∈ [[2, d-1]] such that 2+2m = d. In particular, m 2 ≡ 1 mod d, so g = h m ∈ ⟨h⟩, contradiction!
(2) Else, d = 15, 16, 20, or 24. There is an integer u ′ coprime to d such that, in appropriate coordinates,

M (g) = diag(1 n-4 , ζ d , ζ d a , ζ d b , ζ d c ) M (h u ′ ) = diag(1 n-4 , ζ d , ζ d σ(a) , ζ d σ(b) , ζ d σ(c) )
for some distinct integers a, b, c ∈ [[2, d -1]] coprime to d, and permutation σ of {a, b, c}. If σ = id, then g = h u ′ ∈ ⟨h⟩, contradiction! Nevertheless, let us prove that σ = id. Note that

(h u ′ -u ) α = (h u ′ g -1 ) α (h -u g) α = diag(1 n-3 , ζ d (σ(a)-a)α , ζ d (σ(b)-b)α , ζ d (σ(c)-c)α ),
and thus (h u ′ -u ) α fixes a translated abelian variety W ′ ⊃ W + τ of codimension at most 3. By Proposition 10.11, PStab(W ′ ) is trivial, or cyclic and generated by one junior element k of order 3 or 7. In the second case, as k ∈ PStab(W + τ ), k restricts to an automorphism of the fourfold B ′ , which also has h junior of order d ̸ = 3, 6, 7 acting on it. This contradicts Corollary 10.16. Hence, (h u ′ -u ) α ∈ PStab(W ′ ) = {id}, so for any ℓ ∈ {a, b, c}, (σ(ℓ)-ℓ)α is a multiple of d. However, α was chosen so that g α and g have the same number of distinct eigenvalues, i.e., aα, bα, cα are distinct modulo d. In particular, σ(ℓ)α = ℓα modulo d if and only if σ(ℓ) = ℓ. So σ = id, contradiction! 11.2 Ruling out junior elements of order 6 with four non-trivial eigenvalues. In this subsection, we conclude the proof of Proposition 11.1 by ruling out the one remaining type of junior element fixing at least one subvariety of codimension 4, but no subvariety of codimension 3. It is the type of junior element of order 6, and matrix similar to diag(1 n-4 , ω, ω, ω, -1).

Proposition 11.5. Let A be an abelian variety, G a group acting freely in codimension

2 on A such that A/G has a crepant resolution X. Then there is no junior element of G with matrix similar to diag(1 n-4 , ω, ω, ω, -1).

The proof involves general arguments which we will use later, hence we factor it into a general lemma. Lemma 11.6. Let A be an abelian variety of dimension n, G a group acting freely in codimension 2 on A without translations such that A/G has a simply-connected crepant resolution X. Suppose that g ∈ G fixes 0 ∈ A and has order d. Let W be the abelian subvariety of codimension k in A that g fixes pointwise, and denote by G W the subgroup of G generated by

G gen = G gen -1 = {h ∈ G | ∃ τ ∈ A such that h ∈ PStab(W + τ )} . Then (1)
There is an M (G W )-stable complementary abelian subvariety B to W , which induces a representation ρ :

G W → Aut(B, 0) by ρ(h) := M (h)| B .
(2) If we denote by pr W , pr B the projections induced by the splitting of the tangent space, then, for any h ∈ G W ,

• M (h) = pr W + ρ(h)pr B • pr W (T (h)) = 0, i.e., T (h) ∈ B (3)
The representation ρ is faithful and takes values in SL(H 0 (T B )).

For [START_REF] Auffarth | Smooth quotients of abelian surfaces by finite groups that fix the origin[END_REF], we use that h commutes with g α by ( 6), that g(0) = 0, that T (h) ∈ B by Item 2 of (2), and that

g α | B = -id B . It yields 0 = g α (h(0)) -h(g α (0)) = g α (T (h)) -T (h) = -2T (h), so T (h) is of 2-torsion.
For [START_REF] Bakker | Algebraic approximation and the decomposition theorem for Kähler Calabi-Yau varieties[END_REF], assume that h fixes a point τ in A and has even order. For some β, h β has order 2, and thus equals g α : So, every fixed point of h is a fixed point of g α . The points fixed by g α are all of the form w + τ , with w ∈ W , and τ ∈ B a 2-torsion point. But such a point w + τ being fixed by h, we have that W + w + τ = W + τ is pointwise fixed by h, and in particular, the 2-torsion point τ ∈ B is a fixed point of h.

For [START_REF] Balaji | Holonomy groups of stable vector bundles[END_REF], assume that h is a junior element of order 3. By [START_REF] Andreatta | On the Kummer construction[END_REF], it fixes a point τ ∈ B, and a translated abelian subvariety W ′ + τ , where W ′ is an abelian subvariety of codimension 3 in A. Let B ′ be a ⟨h⟩-stable complementary to

W ′ ∩ B in B. We write τ = w ′ + b ′ , with w ′ ∈ W ′ ∩ B and b ′ ∈ B ′ : It gives h(b ′ ) = h(τ -w ′ ) = τ -w ′ = b ′ , i.e., h fixes b ′ ∈ B ′ . Moreover, since h| B ′ = jid B ′ , it holds 0 = h(b ′ ) -b ′ = (j -1)b ′ + T (h). Multiplying by 2(j 2 -1), we see that 3b ′ is a point of 2-torsion of B ′ . Since h(b ′ ) = b ′ and 3T (h) = T (h), this point 3b ′ is fixed by h.
We can now come back to our Proposition.

Proof of Proposition 11.5. By Remark 11.4, we can assume that G contains no translation other than id A . By contradiction, suppose that there is an element g ∈ G such that g(0) = 0 and, in some coordinates,

M (g) = diag(1 n-4 , ω, ω, ω, -1).
We import the notations of Lemma 11.6, whose hypotheses are satisfied by g for k = 4, d = 6, α = 3. The proof of the proposition now goes in three steps. First, we show that every element of ρ(G W ) is similar to an element of ⟨ρ(g)⟩ ≃ ⟨diag(ω, ω, ω, -1)⟩. Second, we deduce that G W = ⟨g⟩. Third, we use global considerations on fixed loci to derive a contradiction from this description of G W .

Step 1: By Lemma 11.6 (1) and ( 4), there is a G W -stable complementary B to W . As ρ(g) acts on it, B is isogenous to E × E j 3 for some elliptic curve E. By Proposition 10.11, for any τ in A, the group PStab(W + τ ) is trivial, or cyclic generated by one junior element k, and by Corollary 10.16, ρ(k) is similar to ρ(g) (if of order 6) or to ρ(g 2 ) (if of order 3) in GL(H 0 (T B )). By Lemma 11.6 Item 1 of (2), M (k) is therefore similar to M (g) or M (g 2 ) in {id W } × GL(H 0 (T B )). As g 3 commutes with such conjugation matrices, any element of ⟨k⟩ ∪ ⟨g 3 k⟩ = PStab(W + τ ) ∪ g 3 PStab(W + τ ) is similar to a power of g. Now, assume that h ∈ G W is not similar to a power of g. Then Lemma 11.6 (6) shows that 1 and -1 are eigenvalues of ρ(h). Applying Lemma 11.6 (6) again to h 2 , we see that either h 2 is similar to a power of g, or 1 and -1 are eigenvalues of ρ(h 2 ).

If 1 and -1 are eigenvalues of ρ(h 2 ), ρ(h), which has determinant 1, is similar to diag(1, -1, i, i), or to diag(1, -1, -i, -i). Moreover, ρ(h) defines an automorphism of B, and by [17, Thm.13.2.8, Thm.13.3.2], B must thus be isogenous to S × E i 2 for some abelian surface S. We already know that B is isogenous to E×E j 3 , but this contradicts the uniqueness of the Poincaré decomposition of B up to isogeny [START_REF] Birkenhake | Complex abelian varieties[END_REF]Thm.5.3.7].

Hence, h 2 is similar to a power of g, and as 1 is an eigenvalue of multiplicity at least 2 for it, ρ(h 2 ) = id B . Hence, ρ(h) is similar to diag(1, 1, -1, -1).

We just proved that if h ∈ G W is not similar to a power of g, then ρ(h) is similar to diag(1, 1, -1, -1). However, if ρ(h) is similar to diag(1, 1, -1, -1), then ρ(hg) has ω and -ω as eigenvalues, and thus is neither similar to a power of g, nor to diag(1, 1, -1, -1), contradiction. This concludes Step 1.

CHAPTER 12

THE ISOGENY TYPE OF A

This section proves the first part of Theorem 7.6, namely the following proposition, inspired by [START_REF] Oguiso | On algebraic fiber space structures on a Calabi-Yau 3-fold[END_REF]Proof of Lem.3.4].

Proposition 12.1. Let A be an abelian variety of dimension n, G be a finite group acting freely in codimension 2 on A. Suppose that A/G has a crepant resolution X which is a Calabi-Yau manifold. Then either A is isogenous to E j n and G is generated by junior elements of order 3 and 6, or A is isogenous to E u 7 n and G is generated by junior elements of order 7.

Proof. By the M (G)-equivariant Poincaré's complete reducibility theorem [17, Thm.13.5.2, Prop.13.5.4, and the paragraph before], there are M (G)-stable abelian subvarieties Y 1 , . . . , Y s of A such that:

(1) For any i ∈ [[1, s]], Y i is isogenous to a power of a M (G)-stable M (G)-simple
abelian subvariety of A. In particular, by [17, Prop.13.5.5], there is a simple abelian subvariety Z i of Y i such that Y i is isogenous to a power of Z i .

(2) For each i ̸ = j, the set of M (G)-equivariant homomorphisms satisfies

Hom M (G) (Y i , Y j ) = {0}.
(3) The addition map

Y 1 × . . . × Y s → A is an M (G)-equivariant isogeny.
We define

Y I = i∈I Y i , where I = {i ∈ [[1, s]] | Z i ∼ E j } Y J = j∈J Y j , where J = {j ∈ [[1, s]] | Z j ∼ E u 7 } Y K = k∈K Y k , where K = [[1, s]] \ (I ∪ J). The action of M (G) on Y I × Y J × Y K is diagonal by (2)
, and there is a proper surjective finite morphism

A/M (G) → Y I /M (G) × Y J /M (G) × Y K /M (G)

induced by the G-equivariant addition by (3). Composing with projections, we get proper surjective morphisms

f I , f J , f K from A/M (G) to Y I /M (G), to Y J /M (G), and to Y K /M (G).
Let g ∈ G be a junior element. By Propositions 11.3 and 11.5, g has order 3, or 7, or 6 and then five or six non-trivial eigenvalues. By Proposition 10.6, A thus contains an abelian subvariety isogenous to E j 3 , or to E u 7 3 . Hence, dim Y I + dim Y J ≥ 3, so one of the two quotients Y I /M (G), Y J /M (G) has positive dimension. Moreover, by Proposition 10.6 again, if g has order 3 or 6, M (g) acts trivially on Y J and Y K , and if g as order 7, it acts trivially on Y I and Y K . Hence, M (g) acts with determinant 1 on each of the three factors.

But G is generated by its junior elements by Lemma 8.6 and Proposition 8.4. By [START_REF] Khinich | On the Gorenstein porperty of the ring of invariants of a Gorenstein ring[END_REF][START_REF] Watanabe | Certain invariant subrings are Gorenstein. I[END_REF], Y I /M (G), Y J /M (G) and Y K /M (G) are thus normal Gorenstein varieties.

We can now pullback the volume form of Y I /M (G) if it has positive dimension y I = y, of Y J /M (G) of dimension y J = y else, to an M (G)-invariant non-zero global holomorphic y-form on A. Note that the sections of Ω • A are invariant by translations of A, so that we in fact have a G-invariant non-zero global holomorphic y-form on A. It pulls back to X, which is a Calabi-Yau variety. Hence y = n, and either

A ∼ E j n or A ∼ E u 7
n . The order of junior elements generating G is given accordingly by Propositions 10.6, 11.3.

CHAPTER 13

JUNIOR ELEMENTS AND POINTWISE STABILIZERS IN CODIMENSION 5

In this section, we extend the results of Chapters 10 and 11 to codimension k = 5.

In the first subsection, we exclude the one type of junior element with exactly five non-trivial eigenvalues. In the second subsection, we prove the following result.

Proposition 13.1. Let A be an abelian variety on which a finite group G acts freely in codimension 2. Suppose that A/G has a crepant resolution X. Let W be a translated abelian subvariety of codimension k ≤ 5 in A such that {1} ̸ = PStab(W ) < G. Then PStab(W ) is a cyclic group, generated by one junior element g of order 3 or 7.

Ruling out junior elements of order 6 with five non-trivial eigenvalues. Proposition 13.2. Let A be an abelian variety, G a group acting freely in codimension

2 on A such that A/G has a crepant resolution X. Then there is no junior element of G whose matrix is similar to diag(1 n-5 , ω, ω, ω, ω, j).

Proof. Suppose by contradiction that there is an element g ∈ G such that g(0) = 0 and, in some coordinates,

M (g) = diag(1 n-5 , ω, ω, ω, ω, j).
Then there is an abelian subvariety W of codimension 4 in A which is pointwise fixed by g 3 . By Proposition 10.11, PStab(W ) is cyclic, generated by one junior element h. As g 3 ∈ ⟨h⟩, h has even order. However, by Propositions 11.3 and 11.5, it must have order 3 or 7, contradiction! 13.2 The pointwise stabilizer for loci of codimension 5. For proving Proposition 13.1, it is enough to establish the following result.

Proposition 13.3. Let B be an abelian fivefold isogenous to either E j

5 or E u 7 5 , and let p = 3 in the first case, p = 7 in the second case. Let F be a finite subgroup of Aut(B, 0) generated by junior elements of order p, and such that any subgroup of it acting not freely in codimension 4 is cyclic and generated by one junior element of order p. Then F is itself cyclic. Proof of Proposition 13.1 admitting Proposition 13.3. Let W be a translated abelian subvariety of codimension k ≤ 5 in A such that {1} ̸ = PStab(W ) < G. Propositions If p = 7, this yields that gh has four or five eigenvalues of order 7, and thus the characteristic polynomial χ gh⊕gh has exactly eight or ten common roots with Φ 7 , which contradicts its rationality (Lemma 2.76).

If p = 3, the elements of order p in F are each similar to one of the following:

diag(1, 1, j, j, j), diag(1, 1, j 2 , j 2 , j 2 ), diag(j, j, j, j, j 2 ), diag(j, j 2 , j 2 , j 2 , j 2 ).

Most importantly, diag(1, j, j, j 2 , j 2 ) is forbidden because it is neither a power of a junior element, nor acting freely in codimension 4. Let χ be the character of the representation ⟨g, h⟩ ⊂ Aut(B, 0), and a be the number of elements of ⟨g, h⟩ similar to diag(1, 1, j, j, j). As ⟨g, h⟩ ≃ Z 3 × Z 3 , it then has 4 -a elements similar to diag(j, j, j, j, j 2 ). Hence,

0 = ⟨χ, 1⟩ = χ(id) + a(2 + 3j + 2 + 3j 2 ) + (4 -a)(4j + j 2 + 4j 2 + j) = -15 + 6a, contradiction!
Hence, ⟨g⟩ is the only cyclic subgroup of order p in F , so by [170, 5.3.6], F is cyclic.

14.1

The pointwise stabilizers for loci of codimension 6. For proving Proposition 14.1, it is enough to establish the following result. Proposition 14.3. Let B be an abelian sixfold isogenous to either E j 6 or E u 7 6 , and let p = 3 in the first case, p = 7 in the second case. Let F be a finite subgroup of Aut(B, 0) generated by junior elements of order p, such that any subgroup of it acting not freely in codimension 5 is cyclic and generated by one junior element of order p. Suppose that ωid B ̸ ∈ F . Then F is one of the following.

• A cyclic group generated by one junior element of order p.

• An abelian group generated by two junior elements g and h of order p satisfying

E 1 (g) ∩ E 1 (h) = H 0 (W, T W ).
• SL 2 (F 3 ), and the representation M : PStab(W ) → Aut(B, 0) decomposes as σ ⊕3 , where σ is the unique irreducible 2-dimensional faithful representation of SL 2 (F 3 ) over the splitting field Q[j]. In this case, p = 3.

Proof of Proposition 14.1 admitting Proposition 14.3. Let W be a translated abelian subvariety of codimension k ≤ 6 in A such that {1} ̸ = PStab(W ) < G contains no junior element of type diag(1 n-6 , ω, ω, ω, ω, ω, ω). Proposition 13.1 settles the cases when k ≤ 5, so we can assume k = 6. Up to conjugating the whole group G by a translation, we can assume that 0 ∈ W , and apply Proposition 10.3 to obtain a PStab(W )-stable complementary abelian sixfold B to W . By Proposition 12.1 and as an abelian subvariety of A, B is isogenous to either E j 6 or E u 7 6 . Let F = PStab(W ) ⊂ Aut(B, 0). It is generated by junior elements by Proposition 10.3 (3), which have order 3 or 7 by Propositions 11.1, 11.5, 13.2 and since, by assumption, ωid B ̸ ∈ F . Let F ′ be a subgroup of F acting not freely in codimension 5: then there is an abelian variety W ′ ⊋ W of codimension at most 5 such that F ′ ⊂ PStab(W ′ ). By Proposition 13.1, PStab(W ′ ) is cyclic of prime order, so F ′ = PStab(W ′ ) is cyclic generated by one junior element of order 3 or 7.

So Proposition 14.3 applies, and proves Proposition 14.1.

To establish Proposition 14.3, we need numerous lemmas.

Lemma 14.4. Let B be an abelian sixfold isogenous to either E j 6 or E u 7 6 , and let p = 3 in the first case, p = 7 in the second case. Let g ∈ Aut(B, 0) be an element of prime order q. Suppose that, in case ⟨g⟩ acts non-freely in codimension 5, it is cyclic generated by one junior element of order p. We have q ∈ {2, 3, 7}.

Proof. If 1 is an eigenvalue of g, then g has order q = p, as wished.

Suppose that 1 is not an eigenvalue of g. By Lemma 2.76, the characteristic polynomial χ g⊕g is thus a power of Φ q , so q -1 divides 12, so q ∈ {2, 3, 5, 7, 13}.

• If q = 13, then Φ 13 = χ g χ g . But by [START_REF] Washington | Introduction to cyclotomic fields[END_REF]Prop.2.4], Φ 13 is irreducible over Q [j] and

Q[ζ 7 ] ⊃ Q[u 7 ], contradiction.
• If q = 5, then Φ 3 5 = χ g χ g . But by [START_REF] Washington | Introduction to cyclotomic fields[END_REF]Prop.2.4], the cyclotomic polynomial Φ 5 is irreducible over Q

[j] and Q[ζ 7 ] ⊃ Q[u 7 ], contradiction.
Let us describe the 2-, 3-, and 7-Sylow subgroups of F .

Proof. Let g ∈ Z(S) be an element of order p. If ⟨g⟩ is the only subgroup of order p in S, then by [170, 5.3.6], S is cyclic. Control on its order follows from Lemma 14.6. Else, let [h], [k] ∈ S/⟨g⟩ have order p, [h] belonging to the center of this p-group.

Let us prove that

⟨[h]⟩ = ⟨[k]⟩.
If it is the case, then by [170, 5.3.6] again, S/⟨g⟩ is cyclic. A fortiori, S/Z(S) is cyclic, so S is abelian, and S ≃ ⟨g⟩ × C for a cyclic group C containing ⟨h⟩. Control on the factors' orders follows from Lemma 14.6, and then concludes the proof.

If p = 7, then g has an eigenvalue ζ of order 7 with corresponding eigenspace E g (ζ) of dimension 1. By Lemma 14.6, h and k have order 7 in S. As g commutes with h and k, we can thus choose h ′ ∈ [h], k ′ ∈ [k] which both have 1 as an eigenvalue on E g (ζ). Hence, the group ⟨h ′ , k ′ ⟩ does not act freely in codimension 5 on B, so it is cyclic generated by one junior element, and ⟨h ′ ⟩ = ⟨k ′ ⟩ as wished.

If p = 3, let us show that jid B ∈ S. By contradiction, suppose that elements of order 3 in S are all similar to one of the following matrices diag(1, 1, 1, j, j, j), diag(1, 1, 1, j 2 , j 2 , j 2 ), diag(j, j, j, j 2 , j 2 , j 2 ). Take s ∈ S \ ⟨g⟩. As g and s commute, a simple computation shows that one of the products gs, g 2 s, gs 2 , g 2 s 2 will not fall under these three similarity classes, contradiction.

Hence, we can take g = jid B . A fortunate consequence of that choice, of Lemma 2.76, and of the fact that matrices in S all have determinant 1 is that g has no cubic root in S, i.e., every element of order 9 in S has a class of order 9 in S/⟨g⟩. Hence, h and k above have order 3. Moreover, recall that hkh -1 k -1 ∈ ⟨g⟩ = ⟨jid B ⟩. If k is conjugated to jk or j 2 k, then 1, j, and j 2 each are eigenvalues of k, contradiction! Hence, hkh -1 = k, i.e., h and k commute. They commute with g as well, and thus we can find some non-trivial elements in [h] and [k] with a common eigenvector of eigenvalue 1. So ⟨

[h]⟩ = ⟨[k]⟩.
Proof of Proposition 14.3. We now run (see Appendix) a GAP search through all groups with such 2, 3, and 7-Sylow subgroups, which have at most an element of order 2, and no element of order 63. Among the ninety-four of them, only Z 7 and Z 7 × Z 7 can be generated by their elements of order 7, whereas

Z 3 , Z 3 × Z 3 , SL 2 (F 3 ), Q 8 ⋊ (Z 7 ⋊ Z 3 ), and Z 3 × (Q 8 ⋊ (Z 7 ⋊ Z 3 )
) can be generated by their elements of order 3. However, it is easy to check that Q 8 ⋊ (Z 7 ⋊ Z 3 ), and Z 3 × (Q 8 ⋊ (Z 7 ⋊ Z 3 )) have elements of order 28, which by Lemma 2.76 and [START_REF] Washington | Introduction to cyclotomic fields[END_REF]Prop.2.4] cannot occur in Aut Q (E j 6 , 0). The representation theoretic description is easily obtained from GAP for SL 2 (F 3 ), and follows from the condition about freeness in codimension 5 for Z 3 × Z 3 and Z 7 × Z 7 .

Ruling out junior elements of order 6 with six non-trivial eigenvalues.

Proposition 14.9. Let A be an abelian variety, G a group acting freely in codimension 2 on A such that A/G has a crepant resolution X. Then there is no junior element of G with matrix similar to diag(1 n-6 , ω, ω, ω, ω, ω, ω).

In order to prove this, we first reduce to a 6-dimensional situation, where a lot of local information is given by Proposition 14.3. Lemma 14.10. Let A be an abelian variety, G a group acting freely in codimension 2 on A without translations such that A/G has a crepant resolution X. Suppose that there is an element g ∈ G such that g(0) = 0, and with matrix similar to diag(1 n-6 , ω, ω, ω, ω, ω, ω). Then there are complementary ⟨g⟩-stable abelian subvarieties B and W in A such that g| B = ωid B and g| W = id W . For any τ ∈ B, it holds PStab(W + τ ) ⊂ PStab(W ), and if τ is a non-zero 2-torsion point of B, we have PStab(W + τ ) ≃ SL 2 (F 3 ).

Proof. The existence of W and B follows from [START_REF] Birkenhake | Complex abelian varieties[END_REF]Thm.13.2.8]. The fact that ωid B ∈ Aut(B, 0) implies that B is isogenous to E j 6 , by Proposition 10.6. By Schur's lemma, there is an M (G)-stable supplementary S to H 0 (T W ) in H 0 (T A ) (which is not necessarily H 0 (T B ), since M (G) is a larger group than PStab(W )).

Let τ ∈ B. Let h ∈ PStab(W + τ ). The matrices of both g 3 and h split into blocks with respect to the decomposition H 0 (T A ) = H 0 (T W ) ⊕ S, so g 3 commutes with h. As the matrices of g and g 3 have the same eigenspaces (with possibly different eigenvalues), the matrices of g and h commute too, and since G contains no translation, g and h commute themselves. In particular, g(T (h)) = T (h). Let us decompose then

T (h) = w + b with w ∈ W , b ∈ B: 0 = g(T (h)) -T (h) = g(w + b) -w -b = g(b) -b = (ω -1)b.
As by [START_REF] Birkenhake | Complex abelian varieties[END_REF]Cor.13.2.4], ωid B has exactly one fixed point on B, namely 0, we have b = 0, i.e., T (h) ∈ W . But h has a fixed point, so T (h) ∈ Im(id A -M (h)). These two constraints yield T (h) = 0, whence h ∈ PStab(W ).

Suppose now that τ is an non-zero 2-torsion point. As g 3 | B = -id B , g 3 fixes τ , i.e., g 3 ∈ PStab(W + τ ). Since G contains no translation and contains g, no element with matrix similar to diag(1 n-6 , ω, ω, ω, ω, ω, ω) belongs to PStab(W + τ ). Proposition 14.3 therefore applies to PStab(W + τ ), implying that it is isomorphic to SL 2 (F 3 ) (as it contains the element g 3 of order 2). Remark 14.11. This notably shows that, if G contains a junior element g of type diag(1 n-6 , ω, ω, ω, ω, ω, ω) such that g(0) = 0, and W is the maximal abelian subvariety of A fixed by g, then the group G W defined in Lemma 11.6 coincides with PStab(W ). This description of the pointwise stabilizers of the translations of W by 2-torsion points yields the following description of the much larger group PStab(W ). Lemma 14.12. Let A be an abelian variety, G a group acting freely in codimension 2 on A without translations such that A/G has a crepant resolution X. Suppose that there is an element g ∈ G such that g(0) = 0, and with matrix similar to diag(1 n-6 , ω, ω, ω, ω, ω, ω). Let B, W be as in Lemma 14.10. Then there is an element h ∈ PStab(W ) of prime order p if and only if p = 2 or 3. Moreover, a 2-Sylow subgroup S 2 of PStab(W ) is isomorphic to Q 8 , and a 3-Sylow subgroup S 3 contains an even number of junior elements (of order 3). The group PStab(W ) contains exactly 260 junior elements.

Proof. The group PStab(W ) contains a unique element g 3 of order 2, so by [170, 5.3.6], its 2-Sylow subgroup S 2 is cyclic or a generalized quaternion group. Moreover, PStab(W ) acts on a complementary abelian variety to W , which is isomorphic to E j 6 by Proposition 10.6, and the only elements of PStab(W ) with 1 as an eigenvalue are powers of junior elements. Hence, PStab(W ) ⊂ SL 6 (Q[j]) has no element of order 8, i.e., S 2 is isomorphic to Z/2Z, Z/4Z, or Q 8 . But by Lemma 14.10, a copy of Q 8 ⊂ SL 2 (F 3 ) embeds in PStab(W ), and therefore S 2 ≃ Q 8 .

The group PStab(W ) contains g 2 , which has order 3. Note that g 2 commutes with all elements of PStabW , and thus belongs to any 3-Sylow subgroup of it, in particular S 3 . Now, the map h ∈ S 3 → g 2 h 2 ∈ S 3 sends a junior element of order 3 to a junior element of order 3, and is a fixed-point-free involution. Hence, S 3 contains an even number of junior elements (of order 3).

We can also count the number of junior elements in PStab(W ) easily: each of them fixes exactly 2 6 -1 non-zero 2-torsion points of B, and every non-zero 2-torsion point of B is fixed by exactly 4 junior elements by Lemma 14.4. Since B has 2 12 -1 non-zero 2-torsion points, the number of junior elements in PStab(W ) is (2 12 -1)•4

2 6 -1 = 260.
At last, let h ∈ PStab(W ) have prime order p. Suppose by contradiction that p ̸ = 2, 3. By Lemma 14.4, we have p = 7, and since SL 6 (Q[j]) has no junior element of order 7, 1 is not an eigenvalue of h. Hence, all six eigenvalues of h have order 7. Note that h acts by conjugation on the set of junior elements of PStab(W ), whose cardinal, which we just computed, is not divisible by 7. Hence, h commutes with a junior element k ∈ PStab(W ), so hk ∈ PStab(W ) has order 21, and three eigenvalues of order 7, three eigenvalues of order 21. By Lemma 2.76, Φ 7 Φ 21 thus divides the characteristic polynomial of hk⊕hk, but they have respective degrees ϕ( 7)+ϕ( 21 Corollary 14.13. Let A be an abelian variety, G a group acting freely in codimension 2 on A without translations such that A/G has a crepant resolution X. Suppose that there is an element g ∈ G such that g(0) = 0, and with matrix similar to diag(1 n-6 , ω, ω, ω, ω, ω, ω). Let B, W be as in Lemma 14.10. Then the group PStab(W ) has exactly four 3-Sylow subgroups S, T , U and V . There is no junior element in the intersection S ∩ T , and thus S contains exactly 65 junior elements of order 3.

Proof. By Lemma 14.12, there is a positive integer β such that

|PStab(W )| = 8 • 3 β .
The number n 3 of 3-Sylow subgroups in PStab(W ) is thus either 1, or 4.

Let τ ̸ = 0 be a 2-torsion point in B. By Lemma 14.10, there are exactly four junior elements s, t, u, v of order 3 of PStab(W ) fixing τ . We can check in the multiplication table of SL 2 (F 3 ) that the product of any two distinct elements of {s, t, u, v} has order 6. Hence, each 3-Sylow subgroup of PStab(W ) contains at most one element of {s, t, u, v}. So n 3 ≥ 4, hence n 3 = 4. Denote by S, T , U , and V the four 3-Sylow subgroups of PStab(W ).

Suppose by contradiction that S ∩ T contains a junior element h (of order 3). Let τ ̸ = 0 be a non-zero 2-torsion point in B fixed by h. Again, there are exactly four junior elements s, t, u, v of order 3 in PStab(W + τ ), and no two of them belong to the same 3-Sylow subgroup of PStab(W ): In particular, t, u, v belong to either U or V , but that is three elements to fit into two 3-Sylow subgroups, contradiction! Finally, the junior elements of S, T , U , V , partition the set of junior elements of PStab(W ). By the second Sylow theorem, these four partitioning pieces are in bijection, so S has 260 4 = 65 junior elements.

Proof of Proposition 14.9. By contradiction, suppose that G contains a junior element g of type diag(1 n-6 , ω, ω, ω, ω, ω, ω). By Remark 11.4, we can assume that G contains no translation other than id A , and up to conjugating the whole group by a translation, we can assume that g(0) = 0. Now, Lemma 14.12 and Corollary 14.13 apply, but since 65 is odd, they contradict one another. Proof. We prove it by contradiction, using global arguments. Consider such an abelian subvariety W , and apply Lemma 11.6, defining the group G W and a G W -stable complementary B to W . The peculiar features of the representation σ ⊕3 : SL 2 (F 3 ) < G W → Aut(B, 0) yield that B is isogenous to E j 6 . Let g ∈ PStab(W ) ≃ SL 2 (F 3 ) be the unique element of order 2. Recall that g| B = -id B .

Step 1: If h ∈ G W fixes no point, then h has even order.

Proof. Indeed, by Lemma 11.6 (6), either hg fixes a point τ , or 1 and -1 are eigenvalues of h. Clearly, h has even order in the second case. In the first case, hg actually is in PStab(W + τ ), and Propositions 14.1, 14.9 yield that PStab(W + τ ) is isomorphic to Z 3 , Z 3 × Z 3 , or SL 2 (F 3 ). So either hg has order 3, in which case h has even order 6, or hg ∈ PStab(W + τ ) ≃ SL 2 (F 3 ) has order 2, 4, or 6. But then, g ∈ PStab(W + τ ) since G W contains no translation. So h ∈ PStab(W + τ ) fixes points, contradiction!

Step 2: If h ∈ G W has prime order p, then p ∈ {2, 3}. Moreover, if p = 3, h is a junior element or has junior square.

Proof. By Step 1, p = 2 if h fixes no point. By Proposition 14.1 in the case B ∼ E j 6 , p ∈ {2, 3} if h fixes a point.

Hence, in the case when p = 3, we have h ∈ PStab(W + τ ) for some τ ∈ A. Apply Proposition 14.1 to PStab(W +τ ). Note that by Proposition 14.9, ωid B does not appear in ρ(G W ), and as g| B = -id B does, jid B does not. In particular, PStab(W + τ ) can not be Item 2 (i.e., Z 3 × Z 3 ) of Proposition 14.1. In the remaining Items 1 and 3 of that proposition, every order 3 element of PStab(W + τ ) is junior or has junior square, and so is h.

Step 3: A 3-Sylow subgroup S of G W is isomorphic to Z 3 , generated by one junior element.

Proof. Let h ∈ S be a non-trivial element. It has odd order, hence it fixes a point by Step 1, and thus it has order 3 by Proposition 14.1. By Step 2, it is thus junior or a square of a junior element.

Let s ∈ Z(S) be non-trivial, hence again (the square of) a junior element. Let us show that h ∈ ⟨s⟩. As h and s commute, either they have the same eigenspace for the eigenvalue 1, in which case h ∈ ⟨s⟩ as wished, or E s| B (1) and E h| B (1) are in direct sum, in which case jid B ∈ ⟨s| B , h| B ⟩, and so ωid B ∈ ρ(G W ), which contradicts Proposition 14.9. Hence, h ∈ ⟨s⟩ and thus S = ⟨s⟩ ≃ Z 3 .

Step 4: If S 2 , S 3 are 2 and 3-Sylow subgroups of G W , then G W = S 2 ⋊ S 3 .

Proof. By Step 3, no two elements of S 3 are conjugated in G W , so N G W (S 3 ) = C G W (S 3 ), and by Burnside's normal complement theorem [170, 10.1.8], there is a normal subgroup N ◁ G W such that G W = N ⋊ S 3 . By Step 2, N is a 2-group, and it is clearly maximal. As it is normal, it is the unique 2-Sylow subgroup of G, so N = S 2 .

Step 5: S 2 has order 2 9 .

Proof. We first count the number of junior elements in G W . By Lemma 11.6 (9), every junior element in G W fixes at least one 2-torsion point in B. Since it acts trivially on a 3-dimensional translated abelian subvariety of B, it fixes precisely 2 6 of the 2-torsion points in B. Each 2-torsion point τ in B is besides fixed by the four junior elements of PStab(W + τ ) ≃ SL 2 (F 3 ) (by Proposition 14.1 and since g of order 2 belongs to PStab(W + τ )). Hence, there are Proof. We first describe the order and trace of elements h ∈ S 2 different from id A and g. By Lemma 2.76, since B ∼ E j 6 , and by [START_REF] Washington | Introduction to cyclotomic fields[END_REF]Prop.2.4], the characteristic polynomial of ρ(h) = M (h| B ) satisfies

χ ρ(h) = (X -1) α (X + 1) β Φ 4 (X) γ Φ 8 (X) δ ,
with α, β, γ, δ ≥ 0, β being even because of the determinant and α + β + 2γ + 4δ = 6 because of the dimension. Hence, α is even too. If αβ = 0, then by Lemma 11.6, there is τ ∈ A such that h ∈ PStab(W + τ ) ∪ gPStab(W + τ ), so by Proposition 14.1, the only possibility for h other than id and g satisfies χ ρ(h) = Φ 4 3 , hence α = β = 0. Else, α and β are positive. So, (α, β, γ, δ) can be (0, 0, 3, 0),(2, 2, 1, 0),(2, 4, 0, 0), or (4, 2, 0, 0). In particular, h has order 2 or 4, with order 4 if and only if Tr(h| B ) = 0, and order 2 if and only if Tr(h| B ) ∈ {-2, 2}.

Decomposing the representation ρ| S 2 into irreducible subrepresentations yields a splitting coefficient u ∈ N such that u|S 2 | = 72 + 4(m 2 -1), where m 2 is the number of elements of order 2 in S 2 . Denoting by m 4 the number of elements of order 4 in S 2 ans using Step 5, we rewrite (u -4) • 2 9 + 4m 4 = 64. So u ≤ 4.

Note that h ∈ G W junior of order 3 acts by conjugation on the set of elements of order 2 of the normal subgroup S 2 , and the only fixed point is the element g ∈ C G W (⟨h⟩). Hence, m 2 -1 is divisible by 3. So u is divisible by 3, and thus u = 3, and m 2 = 6 • 61 + 1, and m 4 = 144.

Step 7: But m 4 ≥ 6 • 2 6 , contradiction! Proof. Let us show that the number of elements of G W of order 4 fixing a point is exactly 6 • 2 6 . By Lemma 11.6 (8), if h ∈ G W has order 4 and fixes a point, then all its 2 6 fixed points in B are 2-torsion points of B. Moreover, by Proposition 14.1, for any τ ∈ B of 2-torsion, PStab(W + τ ) ≃ SL 2 (F 3 ) contains exactly six elements of order 4. Hence the count of 2 12 •6 2 6 = 6 • 2 6 elements of order 4 fixing a point in G W . And with this contradiction ends the proof of Lemma 14.14. Remark 14.15. Local information would not have been enough to rule out SL 2 (F 3 ). Indeed, considering a simply-connected neighborhood U ⊂ C 6 of 0, which is stable by the action of ρ ⊕3 : SL 2 (F 3 ) → SL 6 (Q[j]), the quotient U/SL 2 (F 3 ) admits a crepant resolution. Let us construct it.

Under the action of SL 2 (F 3 ) on C 6 , exactly four 3-dimensional linear subspaces Z 1 , Z 2 , Z 3 , Z 4 have non-trivial point-wise stabilizers ⟨g 1 ⟩, ⟨g 2 ⟩, ⟨g 3 ⟩, ⟨g 4 ⟩ ≃ Z 3 , where g 1 , g 2 , g 3 , g 4 are the four junior elements of SL 2 (F 3 ). Using Macaulay2, a quick computation shows that the blow-up:

ε : B := Bl I Z 1 ∩I Z 2 ∩I Z 3 ∩I Z 4 (C 6 ) → C 6
is a smooth quasiprojective variety with a four-dimensional central fiber ε -1 (0). In particular, B contains exactly four prime exceptional divisors, one above each Z i .

By the universal property of the blow-up, the action of SL 2 (F 3 ) on C 6 lifts to an action on B. The lifted automorphism gi fixes the exceptional divisor ε -1 (Z i ) pointwise: hence, locally, for any x ∈ B, PStab(x) is generated by pseudoreflections. Hence by Chevalley-Shepherd-Todd theorem, the quotient X := B/SL 2 (F 3 ) is smooth.

We are going to prove that the resolution X → C 6 /SL 2 (F 3 ) is crepant. As SL 2 (F 3 ) ⊂ GL 6 (C) has one conjugacy class of junior elements, by Theorem 2.62, there is exactly one crepant divisor above C 6 /SL(2, 3): A smooth resolution must contain this crepant divisor, and is thus crepant if and only if it contains exactly one exceptional divisor. This is clearly the case for X, since the action of Q 8 ⊂ SL 2 (F 3 ) on B is transitive on the set of the four prime exceptional divisors in B. CHAPTER 16 PROOF OF THEOREM 7.5 In this section, we proceed to the proof of Theorem 7.5, which in fact splits into two pieces. The first piece describes a slight generalization of the situation in dimension 3 [START_REF] Oguiso | On the complete classification of Calabi-Yau threefolds of type III 0[END_REF]. It notably gives an alternative proof of [155, Key Claim 2], replacing the discussion on invariant cohomology and topological Euler characteristics inherent to [155, §3] with group theory and a geometric fixed loci argument ruling out the special linear group SL 3 (F 2 ). Theorem 16.1. Let A be an abelian variety on which a finite group G acts freely in codimension 2 without translations. Suppose that A/G has a resolution X which is a Calabi-Yau manifold. Then, for any two junior elements g, h ∈ G such that ⟨g⟩ ̸ = ⟨h⟩, the intersection of eigenspaces

E M (g) (1) ∩ E M (h) (1) has codimension k ̸ = 3 in H 0 (A, T A ).
The second piece is rather specific to dimension 4. Theorem 16.2. Let A be an abelian variety on which a finite group G acts freely in codimension 2 without translations. Suppose that A/G has a resolution X which is a Calabi-Yau manifold. Then, for any two junior elements g, h ∈ G such that ⟨g⟩ ̸ = ⟨h⟩, the intersection of eigenspaces E M (g) (1) ∩ E M (h) (1)

has codimension k ̸ = 4 in H 0 (A, T A ).
Let us show how these two results imply Theorem 7.5.

Proof of Theorem 7.5, using Theorems 16.1,16.2. Suppose by contradiction that A has dimension 4, and that A/G admits a simply-connected crepant resolution X.

Then by [START_REF] Namikawa | A note on symplectic singularities[END_REF]Thm,Cor.1], X can not be holomorphic symplectic. Hence, by the smooth Beauville-Bogomolov decomposition theorem, X is a Calabi-Yau fourfold. Up to replacing A by an isogenous variety, we can assume that G contains no translation.

If G entails two junior elements g, h such that ⟨g⟩ ̸ = ⟨h⟩, then Theorems 16.1 and 16.2 show that the eigenspaces E M (g) (1) and E M (h) (1) are in direct sum. But they are 3-dimensional subspaces of the 4-dimensional vector space H 0 (T A ), contradiction! So G has all of its junior elements contained in ⟨g⟩, and thus by Item 1 in Theorem 7.6, G = ⟨g⟩ and g has order 3 or 7, and admits 1 as an eigenvalue of multiplicity one. Up to conjugating the whole group G by a translation, we can assume g(0) = 0. Let E ⊂ A be the elliptic curve containing 0 and fixed pointwise by g, and B be its ⟨g⟩-stable supplementary. Hence, G acts diagonally on E × B by {id E } × ⟨g| B ⟩, and the addition map E × B → A is a G-equivariant isogeny by [START_REF] Birkenhake | Complex abelian varieties[END_REF]Thm.13.2.8]. The volume form on E thus pulls back to a G-invariant 1-form on A, and thus to a non-zero global holomorphic 1-form on the Calabi-Yau resolution X of A/G, contradiction. Lemma 16.6. Let A be an abelian variety isogenous to E u 7 3 . Let g, h ∈ Aut(A) be two junior elements of order 7 such that ⟨g, h⟩ contains no translation and no non-junior element fixing points, and E M (g) (1) = E M (h) (1). Then ⟨g, h⟩ cannot be isomorphic to SL 3 (F 2 ).

Proof. The multiplication table of SL 3 (F 2 ) shows that C ⟨g,h⟩ (⟨g⟩) = ⟨g⟩ and N ⟨g,h⟩ (⟨g⟩)/C ⟨g,h⟩ (⟨g⟩) ≃ Z 3 .

Take k ∈ N ⟨g,h⟩ (⟨g⟩) of order 3. Denote by W 1 , . . . , W 7 the seven disjoint translated abelian subvarieties of codimension 3 in A that g fixes pointwise. Then

k Ç 7 i=1 W i å = 7 i=1 W i ,
and since 3 and 7 are coprime, there is some 1 ≤ i ≤ 7 such that k(W i ) = W i . Up to conjugating the whole group ⟨g, h⟩, we can assume that 0 ∈ W i . We apply Lemma 11.6 (2) to g, noting that W = W i and k ∈ ⟨g, h⟩ < G W . It shows that for any w ∈ W i , one has k(w) = w + T (k), and pr W i (T (k)) = 0. As k(W i ) = W i , we obtain T (k) = 0, so k has fixed points and order 3. In particular, it is not a power of a junior element, contradiction.

Proof of Proposition 16.4

M (g) (1) ∩ E M (h) (1) cannot have codimension 4 in H 0 (T A ).
Both propositions are proved by classifying matrices of elements in ⟨g, h⟩, and using representation theory to infer contradictory properties of ⟨g, h⟩. We start with one lemma used in the proof of Proposition 16.7. Lemma 16.9. Let A be an abelian variety isogenous to E u 7 n . Let g, h ∈ Aut(A) be two junior elements of order 7 such that ⟨g, h⟩ contains no translation and no nonjunior element fixing points, and E M (g) (1) ∩ E M (h) (1) has codimension at most 4 in H 0 (T A ). Then for every k ∈ ⟨g, h⟩, the trace of M (k) ⊕ M (k) is at least 2n -8, and equals 2n -7 if k is junior of order 7.

Proof. By Maschke's theorem, there is an ⟨M (g), M (h)⟩-stable supplementary S to E M (g) (1) = E M (h) (1) in H 0 (T A ). Consider the faithful representation ρ of ⟨g, h⟩ given by restricting M to S, with character χ.

Let k ∈ ⟨g, h⟩. If k has a fixed point in A, then k is junior of order 7, and it is clear from Proposition 9.2 that the trace of M (k) ⊕ M (k) equals 2n -7. Else, 1 is an eigenvalue of ρ(k), and we check as in Lemma 16.5 that its characteristic polynomial is one of the following:

Φ 1 4 , Φ 1 2 Φ 2 2 , Φ 1 2 Φ 3 , Φ 1 2 Φ 4 , Φ 1 2 Φ 6 , (X 3 -u 7 X 2 + u 7 X -1)Φ 1 , (X 3 -u 7 X 2 + u 7 X -1)Φ 1 .
The consequence is that ρ(k) ⊕ ρ(k) has non-negative trace, which concludes.

From this lemma follows a reduction to codimension 3 that concludes the proof of Proposition 16.7.

Proof of Proposition 16.7. Denote by 1 both the trivial representation of ⟨g, h⟩ and its character. We have

⟨M | ⟨g,h⟩ , 1⟩ = k∈⟨g,h⟩ Tr M (k) = 1 2 k∈⟨g,h⟩ Tr M (k) + Tr M (k) > (n -4)|⟨g, h⟩|,
by Lemma 16.9, the inequality being strict since ⟨g, h⟩ contains at least one junior element of order 7. Hence, 1 has multiplicity at least n -3 as a subrepresentation of M , i.e., E 1 (M (g)) ∩ E 1 (M (h)) has codimension at most 3 in H 0 (T A ).

We now prove an auxiliary lemma for Proposition 16.8.

Lemma 16.10. Let A be an abelian variety isogenous to E j n . Let g, h ∈ Aut(A) be two junior elements of order 3 such that ⟨g, h⟩ contains no translation and no non-junior element fixing points, and E 1 (M (g)) ∩ E 1 (M (h)) has codimension 4 in H 0 (A, T A ). Then each non-trivial element of ⟨g, h⟩ has order 3.

Proof. By Maschke's theorem, there is an ⟨M (g), M (h)⟩-stable supplementary S to E M (g) (1) + E M (h) (1) in H 0 (T A ), and it has dimension 4. Consider the faithful representation ρ of ⟨g, h⟩ given by restricting M to S, with character χ.

Let k ∈ ⟨g, h⟩. If k has a fixed point in A, then k is junior of order 3. Else, 1 is an eigenvalue of ρ(k), and since the intersection E ρ(g) (j) ∩ E ρ(h) (j) has dimension 2, it must be that 1, j, or j 2 is an eigenvalue of multiplicity 2 of ρ(k). By Lemma 2.76, [START_REF] Washington | Introduction to cyclotomic fields[END_REF]Prop.2.4], and as ρ(k) has determinant one, the characteristic polynomial of ρ(k) in Q[j] is one of the following:

Φ 1 4 , Φ 1 2 Φ 2 2 , Φ 1 2 Φ 3 , Φ 1 2 Φ 4 , Φ 1 2 Φ 6 , (X -j) 3 Φ 1 , (X -j 2 ) 3 Φ 1 .
So the order of k is 1, 3, or an even number.

To conclude, it is enough to show that k cannot have order 2. We prove it by contradiction: Suppose that ρ(k) is similar to diag(1, 1, -1, -1). As the eigenspace E ρ(g) (j) is a hyperplane in S, ρ(gk) has j and -j as eigenvalues. In particular, it is not junior and thus it fixes no point. But its characteristic polynomial should be one of the polynomials listed above, contradiction.

Proof of Proposition 16.8. By Lemma 16.10, ρ(⟨g, h⟩) contains id S and elements similar to diag(1, j, j, j), diag(1, j 2 , j 2 , j 2 ), or diag(1, 1, j, j 2 ). (16.1)

INTRODUCTION

Cone Conjecture. To understand the geometry of a smooth projective variety X, studying the Mori cone of curves NE(X) and its dual, the nef cone Nef(X), is central, especially from the viewpoint of the minimal model program (MMP).

An important part of the relationship between the Mori cone and the MMP is captured by the Cone Theorem and the Contraction Theorem. These theorems assert that the K X -negative part of the Mori cone of a smooth projective variety X is rational polyhedral away from the K X -trivial hyperplane, and the extremal rays of the K X -negative part correspond to some morphisms from X, involved in the MMP. In particular, when X is a Fano variety (namely, -K X is ample), the cone Nef(X) is a rational polyhedral cone, and its extremal rays are generated by semiample classes. In general, however, it is difficult to describe the whole Mori cone, or dually the whole nef cone, even under the slightly weaker assumption that -K X is semiample. For instance, if X is the blowup of P 2 at the base points of a general pencil of cubic curves in P 2 , then -K X is semiample but Nef(X) is not rational polyhedral.

When X is K-trivial, we expect nevertheless that some essential parts of the nef cone of X are rational polyhedral, up to the action of Aut(X). A precise statement, known as the Cone Conjecture, was first formulated by Morrison [143] and Kawamata [START_REF] Kawamata | On the cone of divisors of Calabi-Yau fiber spaces[END_REF]. It was later generalized by Totaro [START_REF] Totaro | The cone conjecture for Calabi-Yau pairs in dimension 2[END_REF] to klt Calabi-Yau pairs (X, ∆) (see Section 18.2 for a definition), thus including much more examples, already in dimension 2.

In this work, we study the Cone Conjecture for certain Calabi-Yau pairs. Let us recall the statement of the Cone Conjecture formulated by Totaro in [START_REF] Totaro | The cone conjecture for Calabi-Yau pairs in dimension 2[END_REF]Conjecture 2.1] (in the absolute situation). For a pair (X, ∆), we define

Aut(X, ∆) := {f ∈ Aut(X) | f (supp(∆)) = supp(∆)}.
We also define the nef effective cone Nef e (X) as Nef e (X) := Nef(X) ∩ Eff(X), where Eff(X) is the effective cone of X. An important prediction of the Cone Conjecture to the Minimal Model Program is that the number of Aut(X, ∆)-equivalence classes of faces of the nef effective cone Nef e (X) corresponding to birational contractions or fiber space structures is finite (see e.g. [192, p.243]).

Nef cones of fiber products. The starting point of this work is a decomposition theorem for the nef cone of a fiber product over a curve.

It begins with the following general question. Let W 1 and W 2 be smooth projective varieties and let ϕ 1 : W 1 → B and ϕ 2 : W 2 → B be surjective morphisms with connected fibers over a smooth base B. Assume that the fiber product W := W 1 × B W 2 is smooth. Question 17.2. Let p i : W → W i be the projection. When do we have

p * 1 Nef(W 1 ) + p * 2 Nef(W 2 ) = Nef(W )? (17.1)
As the nef cone Nef(X) of a smooth projective variety X spans the whole space N 1 (X) R of numerical classes of R-divisors, such a decomposition exists only if

p * 1 N 1 (W 1 ) R + p * 2 N 1 (W 2 ) R = N 1 (W ) R . ( 17.2)
We may then ask which fiber products satisfying the decomposition (17.2) also have the decomposition (17.1). When B is a point, a simple application of the projection formula shows that (17.2) implies (17.1). When B is P 1 and the varieties W i are certain rational elliptic surfaces, the decomposition (17.1) was proven in [START_REF] Grassi | Automorphisms and the Kähler cone of certain Calabi-Yau manifolds[END_REF]Proposition 3.1]. We show that the implication (17.2) ⇒ (17.1) continues to hold for an arbitrary fiber product over a curve. Theorem 17.3. For i = 1, 2, let ϕ i : W i → B be a surjective morphism with connected fibers from a smooth projective variety to a smooth projective curve B. Assume that

1 the variety W = W 1 × B W 2 is smooth; 2 we have p * 1 N 1 (W 1 ) R + p * 2 N 1 (W 2 ) R = N 1 (W ) R . Then p * 1 Nef(W 1 ) + p * 2 Nef(W 2 ) = Nef(W ). As a consequence, we also have p * 1 Amp(W 1 ) + p * 2 Amp(W 2 ) = Amp(W ).
In It provides a way of constructing fiber products (over curves) whose nef cones are not rational polyhedral.

One of them is the desingularized Horrocks-Mumford quintics, studied by Borcea in [START_REF] Borcea | On desingularized Horrocks-Mumford quintics[END_REF] (see also [START_REF] Fryers | The movable fan of the Horrocks-Mumford quintic[END_REF]), and the other is the fiber product of two general rational elliptic surfaces with sections over P 1 , investigated by Schoen in [START_REF] Schoen | On fiber products of rational elliptic surfaces with section[END_REF], by Namikawa in [START_REF] Namikawa | On the birational structure of certain Calabi-Yau threefolds[END_REF], by Grassi and Morrison in [START_REF] Grassi | Automorphisms and the Kähler cone of certain Calabi-Yau manifolds[END_REF]. Both examples are of dimension three.

Some partial results for Calabi-Yau manifolds with Picard number two are due to Lazić, Oguiso and Peternell ( [START_REF] Lazić | On the cone conjecture for Calabi-Yau manifolds with Picard number two[END_REF][START_REF] Oguiso | Automorphism groups of Calabi-Yau manifolds of Picard number 2[END_REF]). Further evidence supporting the Cone Conjecture for Calabi-Yau manifolds includes results obtained by Filipazzi-Hacon-Svaldi [START_REF] Filipazzi | Boundedness of elliptic Calabi-Yau threefolds[END_REF], Kawamata [START_REF] Kawamata | On the cone of divisors of Calabi-Yau fiber spaces[END_REF], Li-Zhao [START_REF] Li | On the relative Morrison-Kawamata cone conjecture[END_REF], Oguiso-Peternell [START_REF] Oguiso | Calabi-Yau threefolds with positive second Chern class[END_REF], Oguiso-Sakurai [START_REF] Oguiso | Calabi-Yau threefolds of quotient type[END_REF], Szendrői [189] and Uehara [START_REF] Uehara | Calabi-Yau threefolds with infinitely many divisorial contractions[END_REF]; see also the recent survey [START_REF] Lazić | The Morrison-Kawamata cone conjecture and abundance on Ricci flat manifolds[END_REF].

Let us now mention some known cases where the Cone Conjecture holds for Calabi-Yau pairs with a boundary divisor ∆ ̸ = 0. Totaro proved the Cone Conjecture for arbitrary klt Calabi-Yau pairs of dimension two in [START_REF] Totaro | The cone conjecture for Calabi-Yau pairs in dimension 2[END_REF]. Prendergast-Smith proved the Cone Conjecture for certain rational elliptic threefolds in [START_REF] Prendergast-Smith | The cone conjecture for some rational elliptic threefolds[END_REF]. The Cone Conjecture was also verified for some Calabi-Yau pairs arising from blow-ups of Fano manifolds of index n -1 and n -2 by Coskun and Prendergast-Smith in [START_REF] Coskun | Fano manifolds of index n -1 and the cone conjecture[END_REF][START_REF] Coskun | Fano manifolds of index n-2 and the cone conjecture[END_REF]. We should also notice that for the Calabi-Yau pairs in [START_REF] Prendergast-Smith | The cone conjecture for some rational elliptic threefolds[END_REF][START_REF] Coskun | Fano manifolds of index n -1 and the cone conjecture[END_REF][START_REF] Coskun | Fano manifolds of index n-2 and the cone conjecture[END_REF], the nef cone is rational polyhedral. Kopper verified the Cone Conjecture for Calabi-Yau pairs arising from Hilbert schemes of points on certain rational elliptic surfaces in [START_REF] Kopper | The nef cone of the Hilbert scheme of points on rational elliptic surfaces and the cone conjecture[END_REF]. In this case, the nef cone may admit infinitely many faces (while the dimension of these varieties are always even).

As we said, the fiber product of two general rational elliptic surfaces with sections over P 1 was investigated by Schoen and others [START_REF] Schoen | On fiber products of rational elliptic surfaces with section[END_REF][START_REF] Namikawa | On the birational structure of certain Calabi-Yau threefolds[END_REF][START_REF] Grassi | Automorphisms and the Kähler cone of certain Calabi-Yau manifolds[END_REF]. It recently came back to light as Suzuki considered a certain higher-dimensional generalization of Schoen's construction and studied its arithmetic properties in [START_REF] Suzuki | Higher-dimensional Calabi-Yau varieties with dense sets of rational points[END_REF], and as Sano used similar ideas to construct non-Kähler Calabi-Yau manifolds with arbitrarily large second Betti number in [START_REF] Sano | Examples of non-Kähler Calabi-Yau manifolds with arbitrarily large b 2[END_REF].

Structure of the paper. We prove Theorem 17.3 in Section 19, and Theorem 17.5 in Section 21. A crucial result by Looijenga, together with preliminaries, is recalled in Section 18. Section 20 describes the construction of Schoen pairs in some detail. CHAPTER 18

PRELIMINARIES

18.1 Notation. The group of automorphisms of X is denoted by Aut(X), and acts on N 1 (X) by pullback. This action ρ : Aut(X) → GL(N 1 (X)) linearly extends to N 1 (X) R , preserving the cones Nef e (X) and Nef + (X). The connected component of the identity in Aut(X) is a normal subgroup Aut 0 (X), which acts trivially on N 1 (X) [START_REF] Brion | Notes on automorphism groups of projective varieties[END_REF]Lemma 2.8]. This induces an action of the discrete group of components π 0 Aut(X) := Aut(X)/Aut 0 (X)

on N 1 (X), that we denote by ρ : π 0 Aut(X) → GL(N 1 (X)).

Klt

Calabi-Yau pairs. The definition of a klt Calabi-Yau pair was given in Section 2.2.

Example 18.1. Let X be a smooth projective variety with -K X semiample. Let m be a positive integer m such that -mK X is globally generated. Then we can always define a Calabi-Yau pair (X, ∆ m,X ) by taking

∆ m,X = 1 m ∆ ′ m,X , where ∆ ′ m,X ∈ | -mK X |.
Moreover, if m ≥ 2 and ∆ ′ m,X is general in its linear system, then the associated pair (X, ∆ m,X ) is a klt Calabi-Yau pair.

In this part, we use the following terminology. Definition 18.2. Let X be a smooth projective variety. We say that X is a Calabi-Yau manifold if the canonical line bundle K X is trivial and h i (X, O X ) = 0 for any 0 < i < dim X. If in addition, X is simply-connected, it is called a strict Calabi-Yau manifold.

18.3 Looijenga's result. We will use the following crucial result in this paper. Proposition 18.3. Let X be a normal projective variety and let H ≤ Aut(X) be a subgroup. Assume that there is a rational polyhedral cone Π

⊂ Nef + (X) such that Amp(X) ⊂ H • Π. Then 1 the variety W = W 1 × B W 2 is smooth; 2 it holds p * 1 N 1 (W 1 ) R + p * 2 N 1 (W 2 ) R = N 1 (W ) R , where p i denotes the projection from W onto W i . For i = 1, 2, let H i ≤ Aut(W i /B) be a subgroup. Let H ≤ Aut(W ) be a subgroup containing H 1 ×H 2 .
Assume that there exists a rational polyhedral cone Π i ⊂ Nef + (W i ) such that H i •Π i ⊃ Amp(W i ). Then Nef + (W ) admits a rational polyhedral fundamental domain for the H-action.

Proof. Let Π be the convex hull of p As W is the blowup of Z at D 2 points, we have K

* 1 Π 1 + p * 2 Π 2 . Then Π is a rational polyhedral cone contained in Nef + (W ). Moreover, Amp(W ) ⊂ (H 1 × H 2 ) • Π ⊂ H • Π as p * 1 Amp(W 1 )+p * 2 Amp(W 2 ) = Amp(W )
W 2 = K Z 2 -D 2 > 0. Since -K W is nef, W is a weak del Pezzo surface.
Let us say something about the nef cone of W , in either case of Lemma 20.2. On one hand, let W be a rational elliptic surface. Clearly Nef e (W ) and Nef + (W ) are subcones of Nef(W ). Moreover, by [START_REF] Totaro | The cone conjecture for Calabi-Yau pairs in dimension 2[END_REF]Lemma 4.2], Nef + (W ) ⊂ Nef e (W ). Then in the papers [START_REF] Totaro | Hilbert's 14th problem over finite fields and a conjecture on the cone of curves[END_REF], [192, p.256, 2nd paragraph], Totaro covers the nef cone Nef(W ) by a set of rational polyhedral subcones {Π E : E is a (-1)-curve}. Since they are rational polyhedral, the Π E are subcones of Nef + (W ). Hence Nef(W ) ⊂ Nef + (W ). This proves the equalities Nef e (W ) = Nef + (W ) = Nef(W ).

On the other hand, a weak del Pezzo surface is easily seen to be a log del Pezzo surface (see [START_REF] Massarenti | Dream Spaces, log Fano varieties and moduli spaces of rational curves[END_REF]Proposition 2.6]), hence by the Cone Theorem [112, Theorem 3.7], its nef cone is a rational polyhedral cone spanned by classes of semiample divisors.

When dim W ≥ 3, the cone Nef(W ) is rational polyhedral, spanned by classes of semiample divisors. This is an immediate corollary of Theorem 20.3 below. Moreover, if dim W ≥ 3, or if dim W = 2 and W is a weak del Pezzo surface, the cone Nef(W ) is rational polyhedral, spanned by classes of semiample divisors.

Finally, note that if D ∈ | -K Z |, then by (20.1), a general fiber of ϕ : W → P 1 is a smooth K-trivial variety. If W has dimension 2, it must be an elliptic curve. In general, we can say the following. (20.1). So by adjunction, ω F ≃ O F , and also we have the exact sequence

Lemma 20.5. If D ∈ | -K Z |, then a general fiber F of ϕ : W → P 1 is a Calabi-Yau manifold, that is, ω F ≃ O F and h i (F, O F ) = 0 for 0 < i < dim F . Proof. Since D ∈ | -K Z |, we have O W (F ) ≃ O W (-K W ) by
0 → ω W → O W → O F → 0.
Since W is rationally connected, we have

h dim W -i (W, ω W ) = h i (W, O W ) = 0 for i ≥ 1. Hence h i (F, O F ) = 0 whenever 1 ≤ i ≤ dim W -2 = dim F -1.

The fiber product

X = W 1 × P 1 W 2 .
We are ready to generalize Schoen's construction and obtain Calabi-Yau pairs in arbitrary dimension. For i = 1, 2, let Z i , D i , W i be as in 20.1. We denote by ϕ i : W i → P 1 the associated fibration, and recall that it has a section.

Denoting by S i the locus of singular fibers of ϕ i in P 1 , we assume S 1 ∩ S 2 = ∅. Moreover, if ϕ 1 : W 1 → P 1 and ϕ 2 : W 2 → P 1 are two rational elliptic surfaces with sections, we require that the elliptic curves ϕ -1 1 (t) and ϕ -1 2 (t) are non-isogenous for a general point t ∈ P 1 .

We consider the fiber product over P 1

X = W 1 × P 1 W 2 p 1 v v ϕ p 2 ( ( W 1 ϕ 1 ( ( W 2 . ϕ 2 v v P 1
As S 1 ∩ S 2 = ∅, the variety X is smooth.

One can also regard X as a complete intersection in P 1 ×Z 1 ×Z 2 of two hypersurfaces in the linear systems

|O P 1 (1) ⊠ O Z 1 (D 1 ) ⊠ O Z 2 | and |O P 1 (1) ⊠ O Z 1 ⊠ O Z 2 (D 2 )|, respectively. In particular, O X (-K X ) = (O P 1 ⊠ O Z 1 (-K Z 1 -D 1 ) ⊠ O Z 2 (-K Z 2 -D 2 )) | X (20.2)
is globally generated. Let m ≥ 1 be an integer. As in Example 18.1, we let

∆ m,X = 1 m ∆ ′ m,X , where ∆ ′ m,X ∈ | -mK X |.
Notice that by construction,

dim X = dim Z 1 + dim Z 2 -1,
and

K X + ∆ m,X ∼ Q 0. Thus, the pair (X, ∆ m,X ) is Calabi-Yau, and is klt if m ≥ 2 and ∆ ′ m,X ∈ | -mK X | is general.
Definition 20.6. The pair (X, ∆ m,X ) constructed above is called a Schoen pair. We may also refer to X alone as a Schoen variety. Lemma 20.7. Any Schoen variety X is simply connected.

Proof. The proof is similar to [178, Lemma 1] and [START_REF] Suzuki | Higher-dimensional Calabi-Yau varieties with dense sets of rational points[END_REF]Lemma 2.1].

Let U ⊂ P 1 be the open subset over which the morphism ϕ : X → P 1 is smooth and set V := ϕ -1 (U ). The natural map ϕ| V : V → U is topologically locally trivial. Denote its fiber by F . Since both ϕ 1 and ϕ 2 have sections, ϕ : X → P 1 also admits a section σ : P 1 → X. Consider the commutative diagram

1 / / π 1 (F ) / / π 1 (V ) / / / / π 1 (U ) σ U * r r / / 1 π 1 (X) / / / / π 1 (P 1 ). σ * r r
Here the first row is exact by the homotopy long exact sequence. By a diagram chase and the fact that π 1 (P 1 ) is trivial, it is enough to check that the image of π

1 (F ) in π 1 (X) is trivial. Write F = F 1 × F 2 , where F i is a general fiber of ϕ i : W i → P 1 for i = 1, 2. Since π 1 (F ) = π 1 (F 1 ) × π 1 (F 2 ), it is enough to show that the image of π 1 (F i ) in π 1 (X)
is trivial, which we prove for i = 1.

A section of ϕ 2 : W 2 → P 1 gives rise to a section s of p 1 : X → W 1 . By construction, the homomorphism π

1 (F 1 ) → π 1 (X) is induced by F 1 → W 1 s - → X, thus factors through π 1 (W 1
). Since it is rationally connected, W 1 is simply-connected, and hence the image of π 1 (F 1 ) in π 1 (X) is trivial.

Proposition 20.8. Suppose that D i ∈ | -K Z i | for both i = 1, 2. Then the Schoen variety X is a strict Calabi-Yau manifold; namely, X is simply connected with ω X ≃ O X , and h i (X, O X ) = 0 for all 0 < i < dim X.
Proof. First of all, (20.2) shows that K X is trivial. Since X is simply-connected by Lemma 20.7, it is enough to show that h p (X, O X ) = 0 for 1 < p < dim X. A general fiber of p 2 , i.e., of ϕ 1 is a Calabi-Yau manifold by Lemma 20.5. Lemma 20.9. Let g : X → Y be a surjective morphism between smooth projective varieties. Assume that a general fiber F of g is a Calabi-Yau manifold and that ω X = O X . Then for every integer i > 0, we have

R i g * O X = ® ω Y , if i = dim X -dim Y, 0, otherwise. Proof. Set r := dim X -dim Y. Since R q g * ω X = R q g * O X is reflexive by [104, Theorem 2.1.(i)
] and [105, Corollary 3.9], and since H q (F, O F ) = 0 for all 0 < q < r and dim H q (F, O F ) = 1 for q = 0 or r, we have

R q g * O X = ®
an invertible sheaf, if q = 0 or r, 0, otherwise.

By Grothendieck-Verdier duality [88, Theorem 3.34], we have

Rg * ω X ≃ RHom(Rg * O X , ω Y [-r]).
The Grothendieck spectral sequence then gives

E p,-q 2 := Ext p (R q g * O X , ω Y ) ⇒ R p-q+r g * ω X .
Note that since Hom(•, ω Y ) is contravariant, we have -q instead of q in E p,-q 2 . (see also [START_REF] Huybrechts | Fourier-Mukai transforms in algebraic geometry[END_REF]Example 2.70.ii)]).

So E p,-q 2 ̸ = 0 only if (p, q) = (0, 0) or (0, r), and Lemma 20.9 follows.

Let w i := dim W i . Applying Lemma 20.9 to p 2 : X → W 2 shows that

R j p 2 * ω X =      O W 2 , if j = 0, ω W 2 , if j = dim w 1 -1, 0, otherwise. It follows from [105, Corollary 3.2] that h p (X, ω X ) = h p (W 2 , O W 2 ) + h p-w 1 +1 (W 2 , ω W 2 )
for all 0 ≤ p ≤ dim X. Since W 2 is rationally connected, this is zero unless p = 0 or

w 1 + w 2 -1.
CHAPTER 21

APPLICATION TO THE CONE CONJECTURE

In this section, we prove Theorem 17.5.

We have defined Schoen pairs (X, ∆ m,X ) in Section 20, arising from fiber products

X = W 1 × P 1 W 2 p 1 v v ϕ p 2 ( ( W 1 ϕ 1 ( ( W 2 . ϕ 2 v v P 1
Lemma 21.1. Any line bundle L on a Schoen variety X can be written

L = p * 1 L 1 ⊗ p * 2 L 2 , where L i is a line bundle on W i .
Proof. Let p ∈ P 1 be a general point and let

F i := ϕ -1 i (p) ⊂ W i . Claim 21.2. The map Ψ : Pic(F 1 ) × Pic(F 2 ) → Pic(F 1 × F 2 ) defined by Ψ(L, M ) = L ⊠ M is an isomorphism.
Proof. First suppose that either W 1 or W 2 is not a rational elliptic surface. Since H 1 (F i , O F i ) = 0 for at least one i ∈ {1, 2}, Claim 21.2 follows from [81, Exercise III.12.6].

Assume now that W 1 and W 2 are rational elliptic surfaces. Then F 1 and F 2 are elliptic curves, and we have a short exact sequence of abelian groups [START_REF] Birkenhake | Complex abelian varieties[END_REF]Theorem 11

.5.1] 0 → Pic(F 1 ) × Pic(F 2 ) Ψ -→ Pic(F 1 × F 2 ) → Hom(F 1 , F 2 ) → 0
where Hom(F 1 , F 2 ) is the group of homomorphisms of group varieties F 1 → F 2 . Since p ∈ P 1 is general, the elliptic curves F 1 and F 2 are non-isogenous by our definition of Schoen varieties. Thus Hom(F 1 , F 2 ) = 0, which proves Claim 21.2.

Let L be a line bundle on X. Claim 21.2 implies that

L |ϕ -1 (p) ≃ L |F 1 ×{u} ⊠ L |{v}×F 2 ,
for any points u ∈ F 2 and v ∈ F 1 .

For each i = 1, 2, we choose a section s i : P 1 → W i and let σ i : W i → X be the induced section:

σ 1 (w 1 ) := (w 1 , s 2 (ϕ 1 (w 1 ))) ∈ W 1 × P 1 W 2 ,
and similarly for σ 2 . We have

L |ϕ -1 (p) ≃ L |F 1 ×{s 1 (p)} ⊠ L |{s 2 (p)}×F 2 ≃ (σ * 1 L) |F 1 ⊠ (σ * 2 L) |F 2 ≃ (p * 1 σ * 1 L ⊗ p * 2 σ * 2 L) |ϕ -1 (p) .
Since p ∈ P 1 is general, by [81, Exercise III.12.4]

L ≃ p * 1 σ * 1 L ⊗ p * 2 σ * 2 L ⊗ O X (D)
for some divisor D whose support is contained in a finite union of fibers of ϕ : X → P 1 . Since the subsets S 1 , S 2 parametrizing singular fibers of ϕ 1 and ϕ 2 respectively are disjoint, the subsets paramatrizing reducible fibers are disjoint as well. Hence, an irreducible component R of a fiber of ϕ is of the form p * i R ′ where R ′ is a multiple of an irreducible component of a fiber of ϕ i : W i → P 1 . Applied to the irreducible components of D, that yields that

Pic(W 1 ) × Pic(W 2 ) p * 1 ⊗p * 2 ---→ Pic(X) is surjective. Lemma 21.3. For every D ∈ Nef(X), one can write D = p * 1 D 1 + p * 2 D 2 , where D i ∈ Nef(W i ).
X) = p * 1 Nef + (W 1 ) + p * 2 Nef + (W 2 ) ⊂ Nef + (X), so Nef(X) = Nef + (X). Similarly, we have Nef(X) = Nef e (X). Therefore, Nef(X) = Nef + (X) = Nef e (X).
Define the subgroups H i ≤ Aut(W i ) by

H i = ® Aut(W i /P 1 ), if W i is a rational elliptic surface, {id W i }, otherwise.
Then there exists a rational polyhedral cone Π i ⊂ Nef + (W i ) such that H i • Π i contains Amp(W i ). Indeed, the case where W i is a rational elliptic surface with -K W i semiample follows from [START_REF] Totaro | Hilbert's 14th problem over finite fields and a conjecture on the cone of curves[END_REF]Theorem 8.2], and the other cases follow from Proposition 20.4. We claim that H 1 × H 2 ≤ Aut(X, ∆ m,X ). Indeed, if neither W 1 nor W 2 is a rational elliptic surface, then H 1 × H 2 is trivial by definition. If both W 1 and W 2 are rational elliptic surfaces, then ∆ m,X = 0 and clearly, H 1 × H 2 ≤ Aut(X). Finally, if one of the W i , say W 1 , is a rational elliptic surface, and the other, say W 2 , is not, then O

X (-K X ) ≃ p * 2 O W 2 (-K Z 2 -D 2 )
. Since p 2 is proper surjective with connected fibers, the pullback p * 2 induces an isomorphism

H 0 (X, p * 2 O W 2 (-m(K Z 2 + D 2 ))) ≃ H 0 (W 2 , O W 2 (-m(K Z 2 + D 2 ))). So ∆ m,X = 1 m p * 2 ∆ m,W 2 , for some divisor ∆ m,W 2 ∈ |O W 2 (-m(K Z 2 + D 2 ))|. Since H 2 = {id W 2 } in this case, it follows that ∆ m,X is invariant under H 1 × H 2 .
This proves the claim.

It then follows from Corollary 19.7 that Nef e (X) = Nef + (X) has a rational polyhedral fundamental domain Π for the Aut(X, ∆ m,X )-action.

Remark 21.5. In [START_REF] Grassi | Automorphisms and the Kähler cone of certain Calabi-Yau manifolds[END_REF], the authors verified the Cone Conjecture for a strict Calabi-Yau threefold X = W 1 × P 1 W 2 , where both W i are rational elliptic surfaces with section, each of whose singular fibers is an irreducible rational curve with a node, and two generic fibers are non-isogenous.

Our proof bypasses the identification shown by Namikawa [ Proof. Recall the linear action ρ : Aut(X) → GL(N 1 (X)) and the induced action ρ : π 0 Aut(X) → GL(N 1 (X)) defined in 18.1. We let Aut * (X) = ρ(Aut(X)) = ρ(π 0 Aut(X)). By Theorem 17.5, there exists a rational polyhedral cone Π

⊂ Nef + (X) such that Amp(X) ⊂ Aut(X, ∆ 1,X ) • Π ⊂ Aut * (X) • Π.
Then, from Proposition 18.3, it follows that there is a rational polyhedral fundamental domain for the Aut * (X)-action on Nef + (X) and the group Aut ). Let V be a smooth complex projective variety. Assume that there exists a rational polyhedral fundamental domain for the action of Aut(V ) on Nef + (V ). Then the set of real structures of V is at most finite.

We end this paper with a short discussion on the minimal models of X. In [START_REF] Namikawa | On the birational structure of certain Calabi-Yau threefolds[END_REF], Namikawa proved that the number of minimal models of X modulo isomorphisms (as abstract varieties) is finite when X is a strict Calabi-Yau threefold obtained from a certain fiber product of rational elliptic surfaces. It would be interesting to investigate the general case. Such finiteness is predicted by a birational version of the Cone Conjecture concerning the structure of movable cones of Calabi-Yau varieties. See [START_REF] Morrison | Beyond the Kähler cone[END_REF][START_REF] Kawamata | On the cone of divisors of Calabi-Yau fiber spaces[END_REF][START_REF] Totaro | The cone conjecture for Calabi-Yau pairs in dimension 2[END_REF][START_REF] Lazić | The Morrison-Kawamata cone conjecture and abundance on Ricci flat manifolds[END_REF] for more details.

INTRODUCTION

Positivity notions are numerous in algebraic geometry: a line bundle can be considered positive, e.g., if it is very ample, ample, strictly nef, nef, big, semiample, effective, pseudoeffective... Some of these notions relate: a very ample line bundle is ample, an ample line bundle is strictly nef and big, a strictly nef line bundle (i.e., a line bundle that has positive intersection with any curve) is nef, a nef line bundle and an effective line bundle are pseudoeffective. These positivity notions, as they tremendously matter in algebraic geometry, have been the subject of a lot of work, to which the books by Lazarsfeld [START_REF] Lazarsfeld | Positivity in algebraic geometry[END_REF][START_REF] Lazarsfeld | Positivity in algebraic geometry[END_REF] are a great introduction. Proving new relationships between these various positivity notions is however a rather naive ambition, if not under strong additional assumptions.

From this perspective, the conjecture by Campana and Peternell [START_REF] Campana | Projective manifolds whose tangent bundles are numerically effective[END_REF] is surprising: they predict that, if X is a smooth projective variety, and the anticanonical bundle -K X is strictly nef, then -K X is ample, i.e., X is a Fano manifold. Their conjecture was in fact proven in dimension 2 and 3, by Maeda and Serrano [START_REF] Maeda | A criterion for a smooth surface to be Del Pezzo[END_REF][START_REF] Serrano | Strictly nef divisors and Fano threefolds[END_REF]. As all Fano manifolds are rationally connected [START_REF] Campana | Connexité rationnelle des variétés de Fano[END_REF][START_REF] Kollár | Rational connectedness and boundedness of Fano manifolds[END_REF], an interesting update on the conjecture is the recent proof by Li, Ou and Yang [128, Theorem 1.2] that if X is a smooth projective variety, and the anticanonical bundle -K X is strictly nef, then X is rationally connected. Their proof uses important results on the Albanese map of varieties with nef anticanonical bundle. Such varieties have been extensively studied too [START_REF] Demailly | Compact complex manifolds with numerically effective tangent bundles[END_REF][START_REF] Zhang | On projective manifolds with nef anticanonical bundles[END_REF][START_REF] Peternell | Threefolds with nef anticanonical bundles: Dedicated to the memory of fernando serrano[END_REF][START_REF] Demailly | Structure theorems for compact Kähler manifolds with nef anticanonical bundles[END_REF][START_REF] Cao | Manifolds with nef anticanonical bundle[END_REF][START_REF] Cao | Albanese maps of projective manifolds with nef anticanonical bundles[END_REF][START_REF] Cao | A decomposition theorem for projective manifolds with nef anticanonical bundle[END_REF].

Positivity notions extend to vector bundles [122, Definition 6.1.1] in the following fashion: a vector bundle E is stricly nef if the associated line bundle O P(E) (1) is strictly nef on P(E). Instead of asking about the positivity of the top exterior power of the tangent bundle, -K X = dim(X) T X , it makes sense to ask about the positivity of intermediate exterior powers r T X , for 1 ≤ r ≤ dim(X) -1.

For r = 1, it is known since Mori [START_REF] Mori | Projective manifolds with ample tangent bundle[END_REF] that projective spaces are the only smooth projective varieties with ample tangent bundle. They are also the only smooth projective varieties with strictly nef tangent bundle, by [START_REF] Li | On projective varieties with strictly nef tangent bundles[END_REF]Theorem 1.4]. Varieties with nef tangent bundle are, on the other hand, governed by another conjecture of Campana and Peternell [START_REF] Campana | Projective manifolds whose tangent bundles are numerically effective[END_REF] which has received a lot of attention: see the survey [START_REF] Muñoz | A survey on the Campana-Peternell conjecture[END_REF], and inter alia [START_REF] Campana | Projective manifolds whose tangent bundles are numerically effective[END_REF][START_REF] Demailly | Compact complex manifolds with numerically effective tangent bundles[END_REF][START_REF] Watanabe | Fano 5-folds with nef tangent bundles and Picard numbers greater than one[END_REF][START_REF] Kanemitsu | Fano 5-folds with nef tangent bundles[END_REF][START_REF] Kanemitsu | Fano n-folds with nef tangent bundle and Picard number greater than n -5[END_REF][START_REF] Muñoz | Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle[END_REF][START_REF] Yang | Toric Fano manifolds with nef tangent bundles[END_REF][START_REF] Li | Smooth projective horospherical varieties with nef tangent bundles[END_REF][START_REF] Demailly | Fano manifolds with nef tangent bundles are weakly almost Kähler-Einstein[END_REF][START_REF] Watanabe | Fano manifolds of coindex three admitting nef tangent bundle[END_REF][START_REF] Kanemitsu | Projective varieties with nef tangent bundles in positive characteristics[END_REF].

For r = 2, it has been proven that varieties with ample second exterior power of the tangent bundle are projective spaces and quadric hypersurfaces [START_REF] Cho | Smooth projective varieties with the ample vector bundle 2 T X in any characteristic[END_REF], varieties with strictly nef second exterior power of the tangent bundle alike. Theorem 22.1. [128, Theorem 1.5] Let X be a smooth projective variety of dimension n ≥ 2, such that 2 T X is strictly nef. Then X is isomorphic to the projective space P n , or to a smooth quadric hypersurface Q n . Partial results were obtained under the nef assumption [START_REF] Watanabe | Positivity of the second exterior power of the tangent bundles[END_REF][START_REF] Schmitz | On exterior powers of the tangent bundle on toric varieties[END_REF]. These results lead us to the following questions. Question 1. Let X be a smooth projective variety of dimension n. Suppose that r T X is strictly nef for some integer 1 ≤ r ≤ n. Is X a Fano variety? Question 2. Let X be a smooth projective variety of dimension n. Suppose that r T X is nef for some integer 1 ≤ r < n, and that X is rationally connected. Is X a Fano variety?

Note that an affirmative answer to the second question would imply an affirmative answer to the first question, by [START_REF] Li | On projective varieties with strictly nef tangent bundles[END_REF]Theorem 1.2]. Also note that the second question is answered negatively for r = n, as there are smooth rationally connected threefolds with -K X nef but not semiample [START_REF] Xie | Rationally connected threefolds with nef and bad anticanonical divisor[END_REF]. The first question is answered affirmatively for smooth toric varieties by [START_REF] Schmitz | On exterior powers of the tangent bundle on toric varieties[END_REF]. In this paper, we answer the second question for r = n -1.

Theorem 22.2. Let X be a smooth projective variety of dimension n ≥ 2 such that the vector bundle n-1 T X is nef and X is rationally connected. Then X is a Fano variety.

This theorem is reminiscent of [47, Proposition 3.10], which states a dichotomy for varieties X with nef tangent bundle: either X is a Fano manifold, or χ(X, O X ) = 0. The proof similarly involves Chern classes inequalities and the Hirzebruch-Riemann-Roch formula. Note that, building on this theorem, [START_REF] Watanabe | Positivity of the exterior power of the tangent bundles[END_REF]Proposition 1.4] very recently gave an affirmative answer to Question 2 in general.

Theorem 22.1 is based on the results of [START_REF] Cho | Characterizations of projective space and applications to complex symplectic manifolds[END_REF] and [START_REF] Dedieu | Numerical characterisation of quadrics[END_REF], which instead of the assumption on 2 T X , feature a much weaker assumption on the length of rational curves. In a similar spirit, we provide the following partial characterizations and their corollaries.

Theorem 22.3. Let X be a smooth projective rationally connected variety of dimension n ≥ 4 such that for each rational curve C in X, we have -K X • C ≥ n -1. Then either X ≃ P 2 × P 2 , or X is a Fano variety of Picard rank ρ(X) = 1.

Corollary 22.4. Let X be a smooth projective variety of dimension at least 4 such that the vector bundle 3 T X is strictly nef. Then either X ≃ P 2 × P 2 , or X is a Fano variety of Picard rank ρ(X) = 1. This theorem and corollary are inspired by the author's Master thesis. Let us briefly discuss the case when ρ(X) = 1. We know that, if X is a cubic or a complete intersection of two quadrics in P n , the vector bundle 3 T X is ample. These are two examples of del Pezzo manifolds, i.e. Fano n-folds of Picard rank 1 and of index n -1. However, we do not know whether other del Pezzo manifolds have strictly nef 3 T X , or whether varieties with strictly nef 3 T X are in general del Pezzo manifolds. We can hardly hope for a characterization of Fano manifolds of Picard rank one on which -K X • C ≥ n -1 for every rational curve C, and it is moreover not clear how to use the positivity of 3 T X beyond that length inequality, cf. Lemma 23.1.

Theorem 22.5. Let X be a smooth projective rationally connected variety of dimension n ≥ 6 such that for each rational curve C in X, we have -K X • C ≥ n -2. Then either X is isomorphic to P 3 × P 3 or X is a Fano variety of Picard rank ρ(X) = 1.

Studying the possibilities in dimension 5 by hand yields the following result.

Corollary 22.6. Let X be a smooth projective variety of dimension at least 5 such that the vector bundle 4 T X is strictly nef. Then either X is isomorphic to one of the following Fano varieties P 2 × Q 3 ; P 2 × P 3 ; P(T P 3 ); Bl ℓ (P 5 ) = P(O P 3 ⊕ O P 3 ⊕ O P 3 (1)); P 3 × P 3 or X is a Fano variety of Picard rank ρ(X) = 1.

These two corollaries were to our knowledge unknown even under the stronger, more classical assumption that 3 T X or 4 T X be ample. The proof of both theorems goes by classifying possible Mori contractions for X. A delicate point is that, while we know that our varieties X with ρ(X) ≥ 2 admit one Mori contraction by the Cone Theorem, we need to construct by hand a second Mori contraction, e.g., to control higher-dimensional fibres in case of a first fibred Mori contraction. Depending on circumstances, we use unsplit covering families of deformations of rational curves, and a result by Bonavero, Casagrande and Druel [START_REF] Bonavero | On covering and quasicovering families of curves[END_REF], or, if X has the right dimension, Theorem 22.2, to produce this second Mori contraction.

CHAPTER 24

RESULTS ON n-1 T X

The following lemma is the main step in the proof of Theorem 22.2.

Lemma 24.1. Let X be a projective n-dimensional manifold such that n-1 T X is nef and X is rationally connected. Then -K X is nef and big.

Proof. By [122, Theorem 6.2.12(iv)], the anticanonical bundle -K X is nef. By the Hirzebruch-Riemann-Roch formula, there is a homogeneous polynomial P of degree n in Q[X 1 , . . . , X n ] with grading deg X i = i such that χ(X, O X ) = P (c 1 (X), . . . , c n (X)).

Note that, as n-1 T X = Ω 1 X ⊗ O X (-K X ), and by [63, Remark 3.2.3(b)], we have

c i Ç n-1 T X å = i j=0 (-1) j Ç n -j i -j å c j (X)c 1 (-K X ) i-j . ( * )
Let us show by induction that c i (X) is a rational polynomial in the c j ( n-1 T X ), for 0 ≤ j ≤ i. Indeed, c 1 (X) = 1 n c 1 ( n-1 T X ). Assume now that for some i, for all 0 ≤ j ≤ i, there is a polynomial P j ∈ Q[X 1 , . . . , X j ] such that c j (X) = P j (c 1 ( n-1 T X ), . . . , c j ( n-1 T X )). Then, setting P i+1 (X 1 , . . . , X i+1 ) = (-1) i+1 X i+1 -i j=0 (-1) i+j+1 Ç n -j i + 1 -j å P j (X 1 , . . . , X j )(P 1 (X 1 )) i+1-j , we have c i+1 (X) = P i+1 (c 1 ( n-1 T X ), . . . , c i+1 ( n-1 T X )) by ( * ). This perpetuates the induction.

In particular, we have χ(X, O X ) = P

Ç P 1 Ç c 1 Ç n-1 T X åå , . . . , P n Ç c 1 Ç n-1 T X å , . . . , c n Ç n-1 T X ååå ,
which is a homogeneous polynomial of degree n in c 1 ( n-1 T X ), . . . , c n ( n-1 T X ). Now, if we suppose that -K X is not big, then c 1 ( n-1 T X ) is not big. Thus, [47, Corollary 2.7] implies χ(X, O X ) = 0. But on the other hand, X is rationally connected, so χ(X, O X ) = 1, contradiction. Lemma 24.3. Let X be a projective n-dimensional manifold such that n-1 T X is nef and X is rationally connected. Then -K X is ample.

Proof of Theorem 22.2. By Lemma 24.1, -K X is nef and big. By the base-point-free theorem [START_REF] Debarre | Higher-dimensional algebraic geometry[END_REF]Theorem 7.32], we can find an integer m such that -mK X is globally generated. Let ε : X → Z be the | -mK X |-morphism. Suppose that it is not finite. By [100, Theorem 2], any irreducible component E of the exceptional locus is covered by rational curves that are contracted by ε. Let C be one of them: we have 0 = -K X • C ≥ 2 by Lemma 23.1, contradiction. So -K X is ample.

Since s * T X/Y has non-positive degree, we obtain

-K Y • C ≥ -K X • s(C) ≥ min{-K X • C ′ | C ′ is a rational curve in X}.
Moreover, if there is an equality, then we have -K Y • C = -K X • s(C), and so V ≃ O P 1 (a 1 ) ⊕k .

If there is almost an equality, then -K Y • C = -K X • s(C) or -K Y • C = -K X • s(C) + 1, so V ≃ O P 1 (a 1 ) ⊕k or V ≃ O P 1 (a 1 ) ⊕k-1 ⊕ O P 1 (a 1 + 1).

Proof of Proposition 25.5. By [START_REF] Höring | Mori contractions of maximal length[END_REF]Theorem 1.3], as π : X → Y is an equidimensional fibration with fibres of dimension r -1, and as it is a Mori contraction of length at least r as well, it is a P r-1 -bundle. Let us show that Y is isomorphic to P r-1 . Since X is smooth and a projective bundle over Y , the variety Y is smooth. By Lemma 25.8, any rational curve C in Y satisfies -K Y • C ≥ r. Moreover, X is rationally connected, so Y is too. By [34, Cor.0.4, 1 ⇔ 10], we get Y ≃ P r-1 .

As P r-1 has trivial Brauer group, there is a vector bundle V of rank r on Y such that π identifies with the natural projection P(V ) → P r-1 . Without loss of generality, we can twist V by a line bundle so that deg ∆ V | ∆ ∈ [[0, r -1]] for any line ∆ in P r-1 . Let ∆ be a line in P r-1 . Then -K P r-1 •∆ = r. By the equality case in Lemma 25.8, the restriction V | ∆ is isomorphic to L ⊕r for some line bundle L on ∆. Hence deg L = 0, so L = O ∆ . By [161, Theorem 3.2.1], the vector bundle V is globally trivial. Hence, X ≃ P r-1 × P r-1 .

Fibred Mori contractions for certain fivefolds

The goal in this section is prove the following result. Proposition 25.9. Let X be a smooth projective fivefold such that 4 T X is strictly nef. Suppose that X admits a fibred Mori contraction. Then X is isomorphic to one of the following projective manifolds P 2 × Q 3 ; P 2 × P 3 ; P(T P 3 ); P(O P 3 ⊕ O P 3 ⊕ O P 3 (1)).

We first establish this classification under the simplifying assumption that X has a P 2 -bundle structure, instead of a fibred Mori contraction. Lemma 25.10. Let X be a smooth projective rationally connected fivefold and such that, for any rational curve C ⊂ X, one has -K X • C ≥ 3. Suppose that p : X → Y is a P 2 -bundle. Then Y is a smooth projective variety, and X is isomorphic to one of the following projective manifolds P 2 × Q 3 ; P 2 × P 3 ; P(T P 3 ); P(O P 3 ⊕ O P 3 ⊕ O P 3 (1)).

Among other things, the proof uses the following lemma. Lemma 25.11. Let V be a vector bundle on a smooth quadric hypersurface Q n . If V is trivial on all lines in Q n , then V is trivial.

Proof. Note that by [START_REF] Ermakova | Vector bundles of finite rank on complete intersections of finite codimension in ind-Grassmannians[END_REF]Theorem 7], it is enough to show that for any x, z ∈ Q n , there exists a point y ∈ Q n such that the lines (xy) and (yz) belong to Q n . Intersecting with n-2 hyperplanes, we can reduce to n = 2, in which case Q 2 ≃ P 1 ×P 1 is covered by two family of lines corresponding to the two rulings. Hence, the point y = (pr 1 (x), pr 2 (z)) satisfies our requirement. As equality holds, V is a covering family of rational 1-cycles with dim Locus(V x ) = 2 ≥ 5 -3, so by [18, Theorem 2, Proposition 1(i)], it admits a geometric quotient p : X → Z, that is a fibred Mori contraction, with a general fiber of dimension 2. If a fiber F of p has dimension 3 or more, then since dim Y ≤ 2, π| F cannot be finite. So π contracts at least a curve B contained in F , which is numerically equivalent to a multiple of C as it lies in a V-equivalence class [18, Remark 1], contradiction.

So p is an equidimensional fibred Mori contraction with fibres of dimension 2, of length -K X • C ≥ 3. By [START_REF] Höring | Mori contractions of maximal length[END_REF]Theorem 1.3], the morphism p is a P 2 -bundle.

We are left supposing that X has a fibred Mori contraction π : X → Y with dim(Y ) = 3 that is not a P 2 -bundle. Let us first prove a few generalities about its fibres.

Lemma 25.14. Let X be a smooth projective n-dimensional variety with a fibred Mori contraction π of length n -k + 1 onto a variety Y of dimension k. Then the general fiber is isomorphic to P n-k .

Proof. The general fiber is a smooth variety F of dimension n-k such that -K F •C ≥ n -k + 1 for any rational curve C in F , and -K F is ample. By [START_REF] Cho | Characterizations of projective space and applications to complex symplectic manifolds[END_REF][START_REF] Kebekus | Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron[END_REF], [84, Theorem 2.1], we obtain F ≃ P n-k .

We recall and prove a fact mentioned in [84, 1.C].

Lemma 25.15. Let X be a smooth projective variety of dimension n ≥ 4 with a fibred Mori contraction π of length n -2 onto a threefold Y . Suppose that π is not equidimensional. Then for any irreducible component F of a fiber of π of dimension n -2, the normalization F of F is isomorphic to P n-2 . Proof. By [START_REF] Höring | Mori contractions of maximal length[END_REF]Theorem 1.3], and as Univ n-3 (X/Y ) → Chow n-3 (X/Y ) is a universal family for the (n -3)-cycles of X over Y , there is a commutative diagram:

X ′ π ′ µ ′ ( ( η ′ / / X π ε ′ / / X π Y ′ µ 6 6 η / / Y ε / / Y
where Y is the normalization of the closure of the π-equidimensional locus of Y in Chow n-3 (X/Y ), X is the normalization of the universal family over it, ε ′ is the evaluation map, Y ′ is a resolution of Y , X ′ is the corresponding normalized fibred product, π ′ is a P n-3 bundle. Note that since Y is Q-factorial, the exceptional loci of µ and of ε are unions of surfaces, hence the exceptional locus of µ ′ is a union of P n-3 -bundles on surfaces.

Let F be an irreducible component of dimension n -2 of a fiber of π, let ν : F → F be its normalization. Let Σ ⊂ Y be one of the surfaces that ε contracts onto π(F ), chosen such that Γ := π -1 (Σ) dominates F . Let S be the strict transform of Σ by η, and let P := π ′-1 (S): it is a P n-3 -bundle over S and it dominates Γ. By the universal property of the normalization, we have a map f : P → F , that fits into the following commutative diagram. Let us show that this family satisfies the hypotheses of [START_REF] Höring | Mori contractions of maximal length[END_REF]Theorem 2.1]. First, note that ν * (-K X | F ) is ample. Since there is a line in X ′ numerically equivalent to ℓ that is disjoint from all exceptional divisors of µ ′ , and since ℓ is contracted by π ′ ,

ν * (-K X | F ) • f * ℓ = -K X • µ ′ * ℓ = -K X ′ • ℓ = -K X ′ /Y ′ • ℓ = -K P n-3 • ℓ = n -2.
Since for any rational curve C in F , it holds ν * (-K X | F ) • C ≥ n -2 by assumption, the family V is unsplit. Moreover, it is a covering family, as ν is birational, µ ′ is surjective and the family of deformations of ℓ is covering. Hence, by [108, Proposition IV.2.5], for a general point x ∈ F , dim V = n -2 + dim Locus(V x ) + 1 -3, so we are left to show that dim Locus (V x ) = n -2 to conclude.

Let us take x and y general in F . It suffices to show that the image by µ ′ | P of a certain fiber P n-3 of π ′ | P contains both x and y, since then there is a line through any two points in P n-3 .

Since x is general and Γ dominates F , it holds dim ε ′-1 (x) = dim Γdim F = n -3 + 2 -(n -2) = 1, so there is a one-dimensional family of cycles passing through x, parametrized by a curve in Σ. As there is a finite map Σ → Chow n-3 (F ) (a composition of inclusions and a normalization), this is a non-trivial family of divisors. Hence, it must cover F , in particular there is one divisor passing through y and x. This divisor is dominated by a fiber of π ′ | P , which concludes.

We now use the fact that π is not a P 2 -bundle (in fact, that π is not equidimensional) to construct covering families of rational curves on X. Before that, we prove a simple lemma. Proof. Note that dim(Y ) = 3, by Lemmas 25.12,25.13. By [START_REF] Druel | Erratum to "characterizations of projective spaces and hyperquadrics[END_REF], let C be a minimal free rational curve in the smooth locus Y 0 ⊂ Y of π. Let s be a minimal section over C. Lemma 25.8 yields 4 ≥ -K Y • C ≥ -K X • s(P 1 ).

The family V of deformations of s(P 1 ) is unsplit. Indeed, suppose by contradiction that it is splitting, i.e. that there is a cycle i a i C i ≡ num s(P 1 ), with C i rational curves, a i ≥ 1 integers, and i a i ≥ 2. Then, intersecting with -K X yields 4 ≥ -K X • s(P 1 ) ≥ 6, contradiction. By Lemma 25.18, V therefore is a covering family. By [108, Proposition IV.2.6], it moreover holds dim Locus(V x ) ≥ -K X • s(P 1 ) -1 ≥ 2 = 5 -3, so by [18, Theorem 2, Proposition 1(i)], there is a geometric quotient p : X → Z, that is a fibred Mori contraction, with general fiber of dimension at least -K X •s(P 1 )-1. By Lemma 25.12, we have dim Z ≤ 3 and by Lemma 25.13, we have dim(Z) = 3, or X is a P 2 -bundle over some three-dimensional base. So dim Z = 3, hence -K X • s(P 1 ) = 3. It also follows that s(P 1 ) is an extremal class in the Mori cone, as wished.

Again, X not being a P 2 -bundle over any smooth base, p is not equidimensional by [START_REF] Höring | Mori contractions of maximal length[END_REF]Theorem 1.3], so a variety F of dimension 3 is contained in a fiber of p. By Lemma 25.15, the normalization n : F → F satisfies F ≃ P 3 .

Since π and p are distinct Mori contractions, they contract no common numerical class of curve, in particular π| F : F → Y is finite onto its image, hence finite surjective for dimensional reasons. There is an effective ramification divisor R ∈ Pic(P 3 ) such that -K P 3 = n * π| F * (-K Y ) -R. As F is an irreducible component of a V-equivalence class, and as V is unsplit, F contains a deformation of s(P 1 ). Let C be the lift to F of a deformation of s(P 1 ) that is contained in F . Then -K P 3 • C ≥ 4, and

n * π| F * (-K Y ) • C = -K Y • C ≤ 4. So R • C ≤ 0, but R ∈ Pic(P 3
) is effective, thus ample or trivial, so R is trivial. The finite map π| F • n : P 3 → Y is thus quasiétale. So, its base change P 3 × Y X → X is also quasiétale, as π : X → Y contracts no divisor.

But X is rationally connected, hence simply-connected, and smooth, so P 3 × Y X → X is an isomorphism. Hence π| F • n : P 3 → Y is an isomorphism too.

Since ρ(Y ) = 1, we have ρ(X) = 2. Since Y ≃ P 3 and 4 ≥ -K Y • C, the curve C is a line. Lemma 25.22. Let X be a smooth projective rationally connected fivefold, such that -K X • C ≥ 3 for any rational curve C ⊂ X. Suppose that X has a fibred Mori contraction π : X → Y with dim(Y ) > 0. If X is not a P 2 -bundle over any smooth projective base, then ρ(X) = 2 and X has two distinct fibred Mori contractions onto P 3 , with corresponding extremal rays generated by the minimal sections s(P 1 ), σ(P 1 ) above lines that lie in each P 3 in the smooth locus of the fibration. Moreover, -K X • s(P 1 ) = -K X • σ(P 1 ) = 3.

Proof. Apply Lemma 25.21 twice.

Proof of Proposition 25.9. If X has a P 2 -bundle structure, then Lemma 25.10 concludes. Suppose that X is not a P 2 -bundle. By Lemma 25.22, X admits exactly two fibred Mori contractions π and p, both onto P 3 . Given the intersection number of -K X with both extremal rays, and as π * s(P 1 ) is a line in P 3 and as p * s(P 1 ) = 0, we have Hence, as ρ(X) = 2, and s(P 1 ) and σ(P 1 ) are independent,

ω X * = π * O P 3 (3) ⊗ p * O P 3 (3).
By Theorem 22.2, -K X is ample. So X is a Fano fivefold, and we just showed that it has index 3. By the classification in [START_REF] Wiśniewski | On Fano manifolds of large index[END_REF], X must then be a P 2 -bundle, which is a contradiction.

Divisorial contractions.

Let us classify divisorial Mori contraction of large length.

Proposition 25.23. Let X be a smooth projective rationally connected variety of dimension n such that -K X • C ≥ 3 for every rational curve C. Then X admits no divisorial Mori contraction of length greater or equal to n -1.

Remark 25.24. In particular, the assumptions are fulfilled if there is 1 ≤ r ≤ n -1 such that r T X is strictly nef, by [128, Theorem 1.2] and Lemma 23.1.

The proof uses the following lemma, that excludes some special contractions of length n -1.

Lemma 25.25. Let X be a smooth projective rationally connected variety of dimension n such that -K X • C ≥ 3 for every rational curve C. Then there is no morphism X → Y that is a blow-up of a smooth point in a smooth variety.

Proof of Lemma 25.25. By contradiction, consider such a smooth blow-up:

f : E ⊂ X → p ∈ Y
Note that since X is rationally connected, so Y is too. Let C be a rational curve through p.

Since -f * K Y = -K X + (n -1)E and since no curve is contained in the blownup locus p, the anticanonical divisor -K Y is stricly nef. By bend-and-break [ Proof. Assume that ε : X → Y is a divisorial Mori contraction contracting the exceptional divisor E to a point. Note that as X is rationally connected, there exists a rational curve C that intersects E without being contained in E. In particular, E • C > 0. Among all such curves, let actually C be one such that -K X • C is minimal. Then we claim that the family V of deformations of C is unsplit. Indeed, suppose by contradiction that it is splitting, i.e.,

C ≡ num i a i C i ,
with rational curves C i and coefficients a i ≥ 1 such that a i ≥ 2. Then E • C > 0, so without loss of generality, E • C 1 > 0. In particular, C 1 intersects E and is not contracted by ε, hence not contained in E. Since -K X has positive degree on all rational curves in X, we have -K X • C 1 < -K X • C, which contradicts the minimality of -K X • C. By [108, Proposition IV.2.6.1], for a general x ∈ Locus(V), dim Locus(V) + dim Locus(V x ) ≥ n + n -2 -1.

In particular, dim Locus(V x ) ≥ n -3, and as X is smooth, E is Cartier, hence intersects Locus(V x ) along a subscheme of dimension at least n -4 ≥ 1. Let B be a curve in this intersection. It is contained in E, hence contracted by ε, hence satisfies E • B < 0. On the other hand, it is contained in Locus(V x ), hence is numerically equivalent to a multiple of C by [START_REF] Andreatta | Generalized Mukai conjecture for special Fano varieties[END_REF]Lemma 4.1]. It has to be a positive multiple, as one sees when intersecting with any ample divisor. But E • C > 0, contradiction.

Corollary 25.28. Let X be a smooth projective variety of dimension n ≥ 5, that is rationally connected and such that -K X • C ≥ n -2 for any rational curve C ⊂ X. Suppose that ε : X → Y is a divisorial Mori contraction. Then Y is smooth and ε is the blow-up of a smooth curve in Y . Lemma 25.29. Let X be a smooth projective variety of dimension n ≥ 3, that is rationally connected and such that for some 1 ≤ r ≤ n -1, for any rational curve C ⊂ X, it holds -K X • C ≥ n + 2 -r. If there is a morphism ε : X → Y that is a blow-up of a smooth curve in the smooth variety Y , then r = n -1.

Proof. Consider such a smooth blow-up:

f : E ⊂ X → ℓ ⊂ Y
As X is rationally connected, so is Y . Fix H an ample divisor on Y . Let C ⊂ Y be a rational curve other than ℓ passing through a point p ∈ ℓ, with H • C minimal among the degrees of all rational curves intersecting ℓ other than ℓ. Fix another point q ∈ C \ C ∩ ℓ. By bend-and-break [START_REF] Debarre | Higher-dimensional algebraic geometry[END_REF]Proposition 7.3], as Y is smooth, if -K Y • C ≥ n + 2, then there is a connected non-integral 1-cycle that is a deformation of C passing through p and q. In particular,

k i=1 a i C i ≡ num C,
with rational curves C i such that p ∈ C 1 , q ∈ C i 0 for some i 0 , coefficients a i ≥ 1, and k i=1 a i ≥ 2. As q ̸ ∈ ℓ, we have that C i 0 ̸ = ℓ, so either C 1 ̸ = ℓ, or C 1 = ℓ and k ≥ 2. Intersecting with H, we see that H • C i < H • C for all i, in particular for C 1 . If C 1 ̸ = ℓ, then H • C 1 contradicts the minimality of H • C. If C 1 = ℓ, then k ≥ 2 and by connectedness of the rational cycle, there is a curve C i 1 ̸ = ℓ that intersects C 1 = ℓ. So C i 1 ̸ = ℓ intersects ℓ and contradicts the minimality, as

H • C i 1 < H • C again. So -K Y • C ≤ n + 1.
The strict transform C ′ ⊂ X of C satisfies E •C ′ > 0. Since K X = f * K Y +(n-2)E, and by assumption,

n + 2 -r ≤ -K X • C ′ ≤ -K Y • C -(n -2) ≤ 3, so r = n -1.
Proposition 25.30. Let X be a smooth projective variety of dimension n ≥ 5, that is rationally connected and such that 4 T X is strictly nef. If there is a morphism ε : X → Y that is a blow-up of a smooth curve in the smooth variety Y , then X is a fivefold and there is a fibred Mori contraction π : X → Z with dim(Z) > 0.

Proof. By Lemma 25.29, we have n = 5. So by Theorem 22.2, -K X is ample. The Mori cone N E(X) is closed, generated by finitely many classes of rational curves. Let E be the exceptional divisor of ε. Note that there exists an extremal ray R = R + [C] of N E(X) on which E • C > 0. Indeed, if there were not such a ray, then E would be non-positive on all curves in X, which is absurd for an effective divisor. So, let R = R + [C] be an extremal ray on which E • C > 0.

Denote the associated Mori contraction by π : X → Z. Since X already had a non-trivial Mori contraction ε, we have dim(Z) > 0. Let us prove that π is a fibred Mori contraction.

By Lemma 25.1, π cannot be a small contraction. Assume by contradiction that it is a divisorial contraction. By Corollary 25.28, the variety Z is smooth and π is a blow-up along a smooth curve of Z. Let E ′ be the π-exceptional divisor. Let ℓ, respectively ℓ ′ , be the image of E, respectively E ′ , in Y , respectively Z. Let F ′ be a general fiber of π| E ′ . It has dimension n -2. Note that F ′ and E intersect, since E • C > 0. Hence, E ∩ F ′ is a subscheme of X of dimension at least n -3. Since ε and π are distinct Mori contractions, the restriction ε| E∩F ′ must be finite onto its image, which is contained in ℓ. So n -3 ≤ 1, contradiction! So π is a fibred Mori contraction.

Proposition 25.31. Let X be a smooth projective variety of dimension n ≥ 5, that is rationally connected and such that 4 T X is strictly nef. If there is a morphism ε : X → Y that is a blow-up of a smooth curve, then Y ≃ P 5 and ε is the blow-up of a line.

Proof. By Proposition 25.30, X is a fivefold and admits a fibred Mori contraction onto a positive dimensional base. So Proposition 25.9 applies, showing that X belongs to a list of certain varieties of Picard number two. Only one of them has a divisorial Mori contraction, namely Bl ℓ (P 5 ) = P(O P 3 ⊕ O P 3 ⊕ O P 3 (1)).

Remark 2 . 72 .

 272 Contrarily to[START_REF] Birkenhake | Complex abelian varieties[END_REF], we do not require automorphisms of A to fix 0 ∈ A. Example 2.73. [17, Example 4.1.13] All elliptic curves are abelian varieties.

Example 2 . 96 .

 296 Consider K = Q[u 16 ] in the notation of Example 2.80. It has degree 2 over K 0 = Q[ √ 2]. Fix η = 4 + 2 √ 2 and ζ = u 16 . Then the first two hypotheses hold. Let us consider the fourth hypothesis. The conjugates of u 16 over Q are -u 16 , v 16 := i 4 -2 √ 2, and -v 16 .

Theorem 2 . 101 . 2 . 2 . 102 .Definition 2 . 103 .

 2101221022103 [START_REF] Suzuki | Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF] Corollary 2, p.94] Let p be a prime number. Let S be a finite p-group of order p s . If a normal abelian subgroup N of S of maximal order has order p n , then s ≤ n(n+1) Remark Obviously, the same inequality holds a fortiori if N is a non-normal abelian subgroup of maximal order.Let us now move on to Sylow theory. Let p be a prime number. Let G be a finite group. A p-Sylow subgroup of G is a maximal p-subgroup of G.The following three theorems often go, by order, under the name of the first, second and third Sylow theorems. Theorem 2.104. [172, Theorem 4.14] Let p be a prime number. Let G be a finite group of order p s m, with p and m coprime. Then there exists a p-Sylow subgroup of G of order p s . Theorem 2.105. [172, Theorem 4.12] Let p be a prime number. Let G be a finite group. Then any two p-Sylow subgroups of G are conjugated.

4. 1

 1 Consequences of Proposition 4.1. We are going to combine Proposition 4.1 with the following corollary of Theorem 2.58.

Lemma 4 . 3 .

 43 Keep the notations. For any closed proper subvariety Z ⊂ P( Ê), for m big and divisible enough and for a general curve Ĉ ∈ p * |mH|, the restricted tautological ζ| Z∩π -1 ( Ĉ) is ample.

Remark 5 . 5 .

 55 has a finite quasiétale cover by an abelian variety by[START_REF] Lu | A characterization of finite quotients of abelian varieties[END_REF] Theorem 1.4], contradiction! This pseudoeffectiveness result can be considered as an interesting improvement of the effectiveness result[START_REF] Greb | Klt varieties with trivial canonical class. Holonomy, differential forms and fundamental groups[END_REF] Theorem 11.1], which says that q(X) = 0 if and only if, for all

CHAPTER 6 THREEFOLDS

 6 IN THEOREMS 3.1 AND 3.2

3 i=0 w i = d 2 together

 32 with the quasismoothness conditions implies that either for all i, gcd(w 0 , . . . , ŵi , . . . , w 3 ) = 1, or d ≡ 2 mod 4 and for all i, gcd(w 0 , . . . , ŵi , . . . , w 3 ) = 1 or 2 helps to apply the criterion. As adjunction formula holds, these general quasismooth wellformed hypersurfaces in weighted projective spaces of dimension 4 have trivial canonical class. Conversely, precisely those general quasismooth wellformed hypersurfaces X d of trivial canonical class in a P(w 0 , . . . , w3 , d2 ) are in the image by f of the N = 3 data. Example 6.18. The wellformed quasismooth Calabi-Yau hypersurface X 1734 in P[START_REF] Im | Rational curves on quotients of abelian varieties by finite groups[END_REF][START_REF] Kamenova | Survey of finiteness results for hyperkähler manifolds[END_REF][START_REF] Kebekus | Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron[END_REF] 578, 867) comes from the N = 3 data (originally denoted n = 4) in[START_REF] Kreuzer | No mirror symmetry in Landau-Ginzburg spectra![END_REF], since 1734 = 2 × 867. The wellformed quasismooth Calabi-Yau hypersurface X 120 in P[START_REF] Andreatta | Generalized Mukai conjecture for special Fano varieties[END_REF][START_REF] Auffarth | Smooth quotients of abelian surfaces by finite groups that fix the origin[END_REF][START_REF] Borcea | Homogeneous vector bundles and families of Calabi-Yau threefolds. II. In Several Complex Variables and Complex Geometry[END_REF][START_REF] Cynk | Generalised Kummer constructions and Weil restrictions[END_REF][START_REF] Dieudonné | Éléments de géométrie algébrique IV. Troisi ème partie[END_REF] comes from the N = 4 data (originally denoted n = 5).

Theorem 7 . 4 .Theorem 7 . 5 .

 7475 Let A be an abelian variety and G be a finite group acting freely in codimension 3 on A. Then A/G has no simply-connected crepant resolution. Let A be an abelian variety and G be a finite group acting freely in codimension 2 on A. If A/G has a simply-connected crepant resolution, then dim(A) ̸ = 4.

  a∈A f (a) := a∈A m(a)f (a).

where 1 A , 1 BNotation 9 . 4 .

 1194 are the indicator functions of A and B. For d ∈ N, we denote by Φ d the d-th cyclotomic polynomial, and by ϕ(d) the degree of Φ d . In other terms, ϕ is the Euler indicator function. For integers a, b, the greatest common divisor of a and b is denoted a ∧ b.

Remark 10 . 5 .

 105 E z := C/Z ⊕ zZ. If z is a quadratic integer, then we denote by Z[z] the Z-algebra that it generates. It holds Z[z] = Z ⊕ zZ ⊂ C. We also define the simple abelian surface S u 16 ,v 16 := C 2 /Z[(u 16 , v 16 )]. Note that the simplicity of S 16 follows from [184, Prop.27].

Proposition 10 . 18 .

 1018 Let B be an abelian fourfold, and let F be a finite subgroup of Aut(B, 0) acting freely in codimension 2, generated by junior elements, containing no junior element of order 3. Then|F | divides 2 4 • 3 • 5 • 7 = 1680.The proof of this proposition relies on the following two lemmas. Lemma 10.19. Let B be an abelian fourfold, and let F be a finite subgroup of Aut(B, 0) acting freely in codimension 2, containing no junior element of order 3. Let p = 3, 5, or 7 divide |F |. Then a p-Sylow subgroup S of F is cyclic of order p.

Lemma 10 . 25 .,

 1025 Let B be an abelian fourfold, and let F be a finite subgroup of Aut(B, 0) acting freely in codimension 2, generated by junior elements. Suppose that p ∈ {3, 5} divides |F |. Let S be a p-Sylow subgroup of F . Then N F (S)/C F (S) is isomorphic to a subgroup of (Z p ) × . Proof. The quotient N F (S)/C F (S) acts faithfully by conjugation on S, and therefore embeds in Aut(S), which by Lemma 10.19 is isomorphic to (Z p ) × . Lemma 10.26. Let B be an abelian fourfold, and let F be a finite subgroup of Aut(B, 0) acting freely in codimension 2, generated by junior elements. Suppose that 5 divides |F |. Let S be a 5-Sylow subgroup of F . Then, if f ∈ N F (S) is a junior element of order 8, [f ] ∈ N F (S)/C F (S) cannot have order 4.Proof. Let f ∈ N F (S) be a junior element of order 8 such that [f ] ∈ N F (S)/C F (S) has order 4, and let g be a generator of S. Looking at the action of f on the eigenspaces of g in coordinates diagonalizing g, with xyzt = -1, and so χ f = X 4 + 1. By Lemma 10.15, no junior element of order 8 has this characteristic polynomial, contradiction.

1 :

 1 .2.5(c)] shows that (g α )| B has strictly more of them than g| B in B. Definition of a certain α ∈ [[0, d -1]] depending on d

  ) = 18 and 12, contradiction! This result has the following consequence.

2 12 ×4 2 6 = 2 8

 62 junior elements in G W . Now, note that by Step 3, the number n 3 of 3-Sylow subgroups of G W equals the number of junior elements in G W . Hence, denoting byS 3 a 3-Sylow subgroup of G W , 3|S 2 | = |G W | = n 3 |N G W (S 3 )| = n 3 |C G W (S 3 )| = 2 9 • 3,since it is easily checked that C G W (S 3 ) = ⟨g, S 3 ⟩ ≃ Z 6 < SL 2 (F 3 ).

Step 6 :

 6 Denote by m 2 , m 4 the number of elements of order 2 and 4 in S 2 . Then m 2 = 6 • 61 + 1 and m 4 = 144.

Conjecture 17 . 1 (

 171 Kawamata-Morrison-Totaro Cone Conjecture). Let (X, ∆) be a klt Calabi-Yau pair. There exists a rational polyhedral cone Π in Nef e (X) which is a fundamental domain for the action of Aut(X, ∆) on Nef e (X), in the sense that Nef e (X) = g∈Aut(X,∆) g * Π, and Π • ∩ (g * Π) • = ∅ unless g * = id.

  by Theorem 17.3. The existence of a rational polyhedral fundamental domain then follows from Proposition 18.3.(1).

Theorem 20 . 3 (

 203 [START_REF] Borcea | Homogeneous vector bundles and families of Calabi-Yau threefolds. II. In Several Complex Variables and Complex Geometry[END_REF] Appendix],[START_REF] Hassett | The weak Lefschetz principle is false for ample cones[END_REF] Theorem 4.3],[START_REF] Beltrametti | A view on extending morphisms from ample divisors[END_REF] Proposition 3.5]). Let Y be a smooth projective variety of dimension ≥ 4 and let j : D → Y be a smooth ample divisor. Suppose that -(K Y + D) is nef. Then Y is a Fano manifold, andj * (Nef(Y )) = Nef(D).In particular, Nef(D) is rational polyhedral, spanned by classes of semiample divisors.In summary, we established the following result. Proposition 20.4. We have Nef e (W ) = Nef + (W ) = Nef(W ).

Proof.

  Lemma 21.3 follows from Lemma 21.1, which by R-linearity, yields the decomposition at the level of N 1 (W ) R , and Theorem 17.3. Theorem 21.4 (= Theorem 17.5). Let (X, ∆ m,X ) be a Schoen pair. Then Nef(X) = Nef + (X) = Nef e (X), and moreover, there exists a rational polyhedral fundamental domain for the action of Aut(X, ∆ m,X ) on Nef e (X). Proof. Since Nef(W i ) = Nef + (W i ) = Nef e (W i ) by Proposition 20.4, we have, by Theorem 17.3 and Lemma 21.3, Nef(

Remark 24 . 2 . 1 Ç= c 1

 24211 If n = 4, we cannot write c 3 (X) as a polynomial in c (X) 3 + 4c 1 (X)c 2 (X), these formulas coming from[90, 4.5.2].

Lemma 2 . 123 ,

 2123 V is unsplit. Fix x ∈ Locus(V) general. By[START_REF] Kollár | Rational curves on algebraic varieties[END_REF] Proposition IV.2.6] and by assumption, we derivedim Locus(V) + dim Locus(V x ) ≥ -K X • C + 5 -1 ≥ 7. So dim Locus(V x ) ≥ 2. By Lemma 2.124, dim Locus(V x ) ≤ dim Y ≤ 2.

5 η

 5 / / Σ ε / / {pt}Let ℓ be a line contained in a fiber of π ′ | P . Let V be the family of deformation of f * ℓ in F .

Definition 25 . 16 .

 2516 Let f : X Y be a rational map. We say that f is almost holomorphic if there is are Zariski open subsets U ⊂ X and V ⊂ Y such that f | U : U → V is a proper holomorphic map. Lemma 25.17. Let f : X Y be an almost holomorphic map. If Y is a curve, then f is holomorphic.

-

  K X • s(P 1 ) = 3 = π * O P 3 (3) • s(P 1 ) = (π * O P 3 (3) ⊗ p * O P 3 (3)) • s(P 1), and similarly -K X • σ(P 1 ) = (π * O P 3 (3) ⊗ p * O P 3 (3)) • σ(P 1 ).

2 . 25 . 26 .Remark 25 . 27 .

 225262527 42, Proposition 3.2] on the smooth variety Y , one can thus assume -K Y• C ≤ n + 1. The strict transform C ′ ⊂ X of C satisfies E • C ′ > 0. Since K X = f * K Y + (n -1)E, we have 3 ≤ -K X • C ′ ≤ -K Y • C -(n -1) ≤ 2, contradiction!Proof ofProposition 25.23. By Ionescu-Wiśnewski inequality, if X admits a divisorial Mori contraction of length ℓ ≥ n -1, the exceptional divisor E and the general fiberF ⊂ E satisfy: dim E + dim F ≥ n + ℓ -1 ≥ 2n -2,i.e., ℓ = n -1 and E = F is contracted onto a point. So [4, Theorem 5.2] applies and shows that this divisorial Mori contraction of X correponds to a blow-up of a smooth point in a smooth variety, which contradicts Lemma 25.25.We now consider divisorial Mori contractions of length n -Proposition Let X be a smooth projective variety of dimension n ≥ 5, that is rationally connected and such that -K X • C ≥ n -2 for any rational curve C ⊂ X.Then X has no divisorial Mori contraction contracting the exceptional divisor to a point. These assumptions are fulfilled if 4 T X is strictly nef, by[128, Theorem 1.2] and Lemma 23.1.

  Proof. Recall[START_REF] Debarre | Higher-dimensional algebraic geometry[END_REF] Proposition 6.10(b)] that the divisorial Mori contraction ε has a unique exceptional divisor E as its exceptional locus. By[111, Lemma 2.62], a ray R + [C] associated to ε satisfies E • C < 0, so such C has negative intersection with at least one effective divisor. Moreover, ε is a Mori contraction of length n -2. So [4, Theorem 5.3] applies, showing that ε either contracts a divisor to a point, or is a blow-up of a smooth curve in a smooth variety Y . By Proposition 25.26, only the latter can occur. Let us finally describe more precisely what happens in the occurrence of Corollary 25.28.

  Given two 1-cycles C 1 and C 2 , we say that they are numerically equivalent and write C 1 ≡ C 2 or C 1 ≡ num C 2 if, for any Cartier divisor D on X, the intersection numbers D • C 1 and D • C 2 equate. Given two Cartier divisors D 1 and D 2 , we say that they are numerically equivalent and write D 1 ≡ D 2 or D 1 ≡ num D 2 if, for any 1-cycle C on X, the intersection numbers D 1 • C and D 2

  Definition 2.10. Let X be a quasiprojective variety. An open set U ⊂ X is called a big open set if its complement has codimension at least two in X. Lemma 2.12. [82, Prop.1.6] Reflexive sheaves on normal quasiprojective varieties are normal. Let f : E → F be a sheaf morphism between two reflexive sheaves on a normal quasiprojective variety X. If there exists a big open set U of X such that f |

	Corollary 2.13.

Definition 2.11. A coherent sheaf E on a quasiprojective variety X is called normal if for every open set U in X, for every big open subset V of U , the restriction map E(U ) → E(V ) is an isomorphism.

  Let U ⊂ X reg be a big open set on which E is locally-free. For m big and divisible enough and for a general element D in |mH|, U ∩ D is a big open set of D. By Bertini theorem and by Lemma 2.14, we can assume D is a normal subvariety of X and E| D is reflexive.

, Lemma 2.3], with the same notations as previously, ζ is pseudoeffective on P if and only if E is pseudoeffective as a reflexive sheaf. Proposition 2.29. Let X be a normal projective variety, H an ample Q-Cartier divisor, E a pseudoeffective reflexive sheaf on X. Then for m big and divisible enough, for a general element D ∈ |mH|, the sheaf E| D is reflexive and pseudoeffective. Proof.

]. Proposition 2.33. Let E be a reflexive sheaf on a normal projective variety X, and

  f : Y → X be a finite dominant morphism of normal projective varieties. Then E is pseudoeffective if and only if f [ * ] E is.

	Definition 2.34. Let D be a Q-Cartier divisor on a normal projective variety X. We
	define its stable base locus
	B(D) :=	Bs(mD),
	m∈M

k general elements of |mH| with k ∈ [[1, n -1]], if we denote by Y the complete intersection D 1 ∩ . . . ∩ D k , E| Y is stable with respect to H| Y . Remark 2.37. Note that the converse is clearly true. Stability a priori weakens through finite Galois reflexive pullbacks: Lemma 2.38. Let

  . . , D p : Y → X be a finite Galois cover of normal projective varieties of dimension n, G its Galois group, H an ample Q-Cartier on X, E be a reflexive sheaf on X. Let F

2.6 Constructions and properties of orbifold Chern classes. Here

  In this set-up, [75, Proposition 3.11.1] grants the additional assumption p * (E| C ) = Ê| Ĉ .

	we recall
	a standard construction for orbifold first and second Chern classes of a reflexive sheaf E
	on a normal projective variety X, whose singularities in codimension 2 are all quotient
	singularities. Note that normal projective klt varieties fall into this framework by the
	classical result [168, Corollary 1.14], [71, Proposition 9.4]. References for this matter
	include [147, 133, 75, 76], [106, Chapter 10, 11].
	We first make plain what we mean by quotient singularities.
	Definition 2.46. Let X be a normal quasiprojective variety. We say that X has a
	quasiétale Q-structure if there is a finite collection of quasiprojective varieties (V α ) α∈A
	together with morphisms

  where i α is the inclusion of an open subvariety, pα is Galois of group H α ◁ G, and p α is Galois of group G/H α ≃ G α .

		finite and acts faithfully freely in codimension
	1 (so that p α is quasiétale), and the union α∈A p ′ α (p α (V α )) covers X.
	Lemma 2.47. [147, Section 2, p.277] Let X be a normal quasiprojective variety
	equipped with a quasiétale Q-structure (V α , G α , p ′ α ) α∈A . Then there is a normal fi-nite Galois cover X of X with group G such that for each α, we have a commutative
	diagram
	Vα	pα / / V α	p α / / V α /G α
	i α		p ′ α
	X	p	/ / X

Definition 2.49. Let

  0 for all ample divisors H 1 , . . . , H n-2 . More is to be said about this example in Section 2.7. x ∈ X be a normal singularity. We define the local fundamental group π loc 1 (X, x) as the group π 1 (B \ {x}), where B is the intersection of X with a small Euclidean ball containing x.

	Example 2.50. Let S be a normal klt surface. Following [106, Definition 10.7], we
	define its orbifold Euler number as
	e orb (S) = e(S) -	p∈S sing	1 -	1 1 (S, p)| |π loc	.
	By [106, Theorem 10.8], we then have ĉ2 (S) = e orb (S).
	2.

7 Some inequalities for (orbifold) Chern classes. The

	first two parts of
	this thesis focus on minimal varieties, i.e. varieties with nef canonical divisor. The
	following inequality relates the first and the second Chern class of a smooth minimal
	variety.
	Theorem 2.51. [140, Theorem 1.1] Let X be a smooth projective variety of dimension
	n, with K X nef. Then for any ample line bundles H 1 , . . . , H n-2 on X,

  an extension of w, then by[211, Ch.VI, Par.12], the set of all extensions of w is exactly {v • g | g ∈ G}. In particular, all extensions of w have the same ramification index. . The local rings H Y,E and H X,F are discrete valuation rings for the valuations v E and v F .If we assume that F dominates E, then p : X → Y induces an injective morphism of local rings p ♯ : H Y,E → H X,F by[START_REF]Stacks Project[END_REF] Lem.29.8.6]. The maximal ideals m E ⊂ H Y,E and m F ⊂ H X,F relate by p ♯ (m E ) = m r F , where r is the ramification index of F over

	Remark 2.65. When considering divisorial valuations, ramification indices and ex-
	tension properties carry a geometrical meaning. Let X, Y be normal complex ana-
	lytic varieties endowed with their sheaves of holomorphic functions H X and H Y . Let
	p : X → Y be a finite Galois morphism of group G, and let E, F be prime divisors in
	X, Y

Proposition 2.67. [211

  , p.77, Cor.] If the residue field R v /m v has characteristic zero, then the inertia group G T is cyclic of order Ram(v, k(Y )).

Proposition 2.68. [94

  , Cor.2.7 and p.11, Par.1] Suppose that U is an open simplyconnected subset of C n , G is a finite subgroup of GL n (C) stabilizing U , and p : U → U/G = Y is the quotient map. Let h ∈ GL n (C) be a junior element. Then:

•

  S l (E| C ) remains H| C -stable of zero slope, by Lemma 2.36; by Lemma 2.40, E| C is H| C -stable of zero slope as well. Apply now Lemma 2.43 and Remark 2.44: they establish that ζ| π-1 ( Ĉ) is nef and that, for any closed proper variety W

  , Corollary 2.41, and Lemma 2.38, but no other property of E. Keep the notations. If ζ is not nef, then there is a closed proper subvariety W of P( Ê) that is not contained in a fiber of π such that, for a very general curve Ĉ ∈ p

	Lemma 4.4.

* |mH|: ζ| W ∩π -1 ( Ĉ) is nef and not big. This result essentially relies on [86, Lemma 3.4]. Proof. Denote by µ : S → Ŝ the minimal resolution of Ŝ, by Ẽ := µ * Ê, by ζ the tautological bundle of P( Ẽ). We have a Cartesian diagram with compatibility of tautological bundles: P( Ẽ) π µ ′ / / P( Ê) π S µ / / Ŝ Note that P( Ẽ) with its tautological ζ is a smooth modification of P( Ê) just as in Definition 2.28. Hence, ζ is pseudoeffective.

We suppose that ζ is not nef. In particular, ζ is pseudoeffective but not

Proof of Theorem 3.3.

  and not big. The first condition shows that π(W ) is not a point. So Lemma 4.3 applies, hence ζ| W ∩π -1 ( Ĉ) is ample, contradiction! As it follows from the discussion in Section 4.1, Theorem 3.3 is halfway. Here is what remains to prove: Let X be a normal projective klt variety of dimension n with an ample Q-Cartier divisor H. Let E be a reflexive sheaf on X, such that:

	CHAPTER 5
	PROOF OF THEOREM 3.3 AND OF THEOREM 3.2
	5.1 Theorem 5.1.

  this vertex is a singular point in P. It gives rise to a singular point in X if and only if it lies in it, ie w i does not divide d.• an edge in P is a line with equation y j = 0 for all j ∈ J, for a certain J ⊂ [[0, 4]] of cardinal 3. If gcd(w j ) j̸ ∈J ̸ = 1, the edge is in P sing . Recall that X is taken general in its linear system. Hence, an edge in P lies entirely in X if and only if (w

j ) j̸ ∈J do not partition d, in X sing if and only if (w j ) j̸ ∈J do not partition d and have a non-trivial common divisor. If an edge in P sing does not lie entirely in X, it gives a finite amount of points in X sing .

  finite of degree d and étale over the big open set q -1 (X reg ) ⊂ C * X . Normalizing, the map C′ → C * X has degree d and is étale over a big open set as well. As C * X is smooth, this map is actually finite étale; by [53, Lemma 3.2.2(ii)], π ét 1

Table 9 .

 9

	1 4 , 3 4

2: Possibilities for u, 1 d A, α, S A,d (u) such that S A,d (u) ≤ 1 Lemma 9.7. Let k ∈ N. For each m ∈ [[1, k]]

, let u m ≥ 2 and d m ≥ 3 be integers, such that u m divides d m , and suppose that there are a positive integer α m and a multiset A m such that:

Table 9 .

 9

	, 8	1,	1 4 , 1 8 , 5 8	✗
		2,	1 4 , 1 4 , 1 8 , 3 8	✗
	4, 12	2,	1 4 , 1 4 , 1 12 , 5 12	✗
	5, 10	1,	1 5 , 2 5 , 1 10 , 3 10	✗
	6		1 6 , 5 6	✗
			1 6 , 1 6 , 1 6 , 1 6 , 1 6 , 1 6	✓
	6, 8	3,	1 6 , 1 6 , 1 6 , 1 8 , 3 8	✗
	6, 12	2,	1 6 , 1 6 , 1 12 , 7 12	✗
		3,	1 6 , 1 6 , 1 6 , 1 12 , 5 12	✗
	7		1 7 , 2 7 , 4 7	✓
	8		1 8 , 1 8 , 3 8 , 3 8	✓
	8, 12	1,	1 8 , 3 8 , 1 12 , 5 12	✗
	12		1 12 , 1 12 , 5 12 , 5 12	✓
	15		1 15 , 2 15 , 4 15 , 8 15	✓
	16		1 16 , 3 16 , 5 16 , 7 16	✓
	20		1 20 , 3 20 , 7 20 , 9 20	✓
	24		1 24 , 5 24 , 7 24 , 11 24	✓

3: Possibilities for k parcels of data u m , α m , 1 dm A m such that k m=1

Table 10

 10 respectively, and Lemma 10.8 is[START_REF] Birkenhake | Complex abelian varieties[END_REF] Cor.13.3.4, Cor.13.3.6]. In the other cases, the computation relies on the same ideas as[START_REF] Birkenhake | Complex abelian varieties[END_REF] Cor.13.3.6], as we will soon see.

	k	S k		C ϕ(k)/2 /Im(f (S k ))
	3	{j}		E j
	4	{i}		E i
	6	{ω}		E j
	7	{ζ 7 , ζ 7	2 , ζ 7	4 }	E u 7	3
	8	{ζ 8 , ζ 3 8 }		E u 8	2
	12	{ζ 12 , ζ 5 12 }	E i	2
	15 {ζ 15 , ζ 2 15 , ζ 4 15 , ζ 8 15 }	E u 15	4
	16 {ζ 16 , ζ 3 16 , ζ 5 16 , ζ 7 16 }	S u 16 ,v 16	2
	20 {ζ 20 , ζ 3 20 , ζ 7 20 , ζ 9 20 }	E u 20	4
	24 {ζ 24 , ζ 5 24 , ζ 7 24 , ζ 11 24 }	E u 24

.2. Remark 10.9. For k = 3, 4, 6, 8, 12, we have S k = {j}, {i}, {j}, {ζ 8 , ζ 8 3 }, and {ζ 12 , ζ 12 5 } 4 Table 10.2: Computing C ϕ(k)/2 /Im(f (S k )) for given S k stemming from a junior element.

Table 10

 10

	.3. For

  g has determinant 1 and ⟨g⟩ acts freely in codimension 2, we work out all possibilities by hand and add them to the table. One example falling in this case is diag(1, ζ 7 , ζ 7 2 , ζ 7 4

  If v 1 does not divide v 2 , then by Lemma 2.76 again, ϕ(v 1 ) + ϕ(v 2 ) ≤ 4.

	of them
	is diag(ζ 5 , ζ 2 5 , ζ 3 5 , ζ 4 5 ).
	(b)

1 , v 2 ). Listing integers of Euler number 4, u ∈ {5, 8, 10, 12}. (a) If v 1 divides v 2 , then v 2 = u. We investigate all possibilities of determinant 1 satisfying Lemma 2.76 by hand and add them to the table. One

  By uniqueness in the Poincaré decomposition of B [17, Thm.5.3.7], B is not isogenous to any of the other special abelian varieties appearing in Lemma 10.15. So, by Lemma 10.15 again, h being junior must have order 7. By Proposition 9.2, any junior element k of order 7 acting on a fourfold with ⟨k⟩ acting freely in codimension 2 are similar to diag(1, ζ 7 , ζ 7 2 , ζ 7 4

h is similar to diag(ζ 12 , ζ 5 12 , ζ 12 , ζ 5 12 ) and B is isomorphic to E i 4 .

Proof. If g has order 7, then by Lemma 10.15, B is isogenous to E × E u 7

3

for some elliptic curve E.

Lemma 10.20. Let

  So gh 2 has eigenvalue ζ 73 on v, and ζ 7 or ζ 73 on w, which in either case contradicts Lemma 10.15. So g(w) = h(w), i.e., gh -1 has eigenvalue 1 with multiplicity two. By freeness in codimension 2, g = h, hence h ∈ ⟨g⟩. B be an abelian fourfold, and let F be a finite subgroup of Aut(B, 0) acting freely in codimension 2. If not trivial, a 2-Sylow subgroup S of F is cyclic or a generalized quaternion group, and its order divides16.

[START_REF] Beltrametti | A view on extending morphisms from ample divisors[END_REF]

, {g(w), h(w)} = {ζ 7 2 w, ζ

7 4

w}.

  We start by showing that, provided F is cyclic, it is generated by one junior element. Let F be a cyclic group. If E is a set of generators of F and all elements of E have the same order, then any element of E actually generates F . Proof. Suppose F = Z d and every element of E has order k dividing d. Then E is actually a subset of Z k ⊂ Z d , and since E must generate Z d , it must be k = d. So any element e ∈ E satisfies ⟨e⟩ = Z d = F . Let B be an abelian fourfold, and let F be a finite subgroup of Aut(B, 0) acting freely in codimension 2, generated by junior elements, containing no junior element of order 3 or 7. If F is cyclic, then F is generated by one junior element. Proof. Assume that F is cyclic. If F contains one junior element of order 8, 15, 16, 20, or 24, then by Corollary 10.16, all junior elements have the same order and we use Lemma 10.23 to conclude.

	Lemma 10.23. Corollary 10.24.

Proposition 10.22. Let B be an abelian fourfold, and let F be a finite subgroup of Aut(B, 0) acting freely in codimension 2, generated by junior elements, containing no junior element of order 3. Suppose that 7 divides |F |. Then F is cyclic and generated by one junior element.

Proof. Let S be a 7-Sylow subgroup of F . By Lemma 10.21, F = N ⋊ S, where N is a normal subgroup of F , and by Proposition 10.18, |N | divides 240. A simple GAP program in the appendix checks that a group of order dividing 240 cannot have an automorphism of order 7. So S acts trivially on N , i.e., F = N ×S. But F is generated by its junior elements, which all have order 7 by Corollary 10.16. So N is trivial, and F = S is cyclic of order 7. Now we can focus on the case when F contains no junior element of order 3 or 7.

Proposition 10.27. Let

  Suppose that a 2-Sylow subgroup of F is trivial or cyclic. Then F is cyclic. Proof of Proposition 10.28. Let us write |F | = 2 α • 3 β • 5 γ with α ∈ [[0, 4]], β, γ ∈ [[0, 1]].By Lemma 10.19 and by assumption, the Sylow subgroups of F are cyclic, so[170,

B be an abelian fourfold, and let F be a finite subgroup of Aut(B, 0) acting freely in codimension 2, generated by junior elements, containing no junior element of order 3 or 7. Then a 2-Sylow subgroup of F is either trivial, or cyclic.

Proposition 10.28. Let B be an abelian fourfold, and let F be a finite subgroup of Aut(B, 0) acting freely in codimension 2, generated by junior elements, containing no junior element of order 3 or 7. pp.290-291] applies and F is a semidirect product:

  by Lemma 10.20.

• |F | divides 240, by Proposition 10.18. • F has no element of order 60 or 40, by Lemma 10.15. • If F has elements of orders o, o ′ ∈ {15, 20, 24}, then o = o ′ , by Lemma 10.15.

  The aim of this section is to rule out eight out of the twelve types of junior elements presented in Proposition 9.2, namely those which fix pointwise at least one subvariety of codimension 4, but no subvariety of codimension 3.

		CHAPTER 11
	RULING OUT JUNIOR ELEMENTS IN CODIMENSION 4
		ζ 18 , ζ 7 18 , ζ 13 18 ) iid	3 8 , ζ 3 8 )⟩,
	12 C 4 /⟨diag(ζ 12 , ζ 12 , ζ 5 diag(ζ 12 , ζ 5 12 , ζ 12 , ζ 5 12 ) 12 , ζ 5 12 )⟩, C 4 /⟨diag(ζ 15 , ζ 2 15 , ζ 4 B ≃ E i 15 , ζ 8 15 )⟩ 4
	20 have a crepant Fujiki-Oka resolution, and by [176, Prop.3.9], diag(ζ 20 , ζ 9 20 , ζ 13 20 , ζ 17 20 )
	diag(1, ζ 7 , ζ 7 16 )⟩, C 4 /⟨diag(ζ 20 , ζ 3 2 , ζ 7 16 , ζ 9 16 , ζ 5 C 4 /⟨diag(ζ 16 , ζ 3 20 , ζ 7 4 ) 20 , ζ 9 20 )⟩, C 4 /⟨diag(ζ 24 , ζ 5 B ∼ E × E u 7 3 24 , ζ 7 24 , ζ 11 15 )⟩
	14 admit no toric crepant resolution. They could nevertheless have a non-toric crepant diag(-1, ζ 14 , ζ 9 14 , ζ 11 14 ) diag(ζ 8 , ζ 3 8 , ζ 8 , ζ 3 8 ) B ≃ E u 8 4 resolution.
	24	diag(ζ 24 , ζ 11 24 , ζ 17 24 , ζ 19 24 )
	15	diag(ζ 15 , ζ 2 15 , ζ 4 15 , ζ 8 15 )	B ≃ E u 15	4
	30	diag(ζ 30 , ζ 17 30 , ζ 19 30 , ζ 23 30 )
	16	diag(ζ 16 , ζ 3 16 , ζ 5 16 , ζ 7 16 )	B ≃ S u 16 ,v 16	2
		diag(ζ 16 , ζ 7 16 , ζ 11 16 , ζ 13 16 )
	20	diag(ζ 20 , ζ 3 20 , ζ 7 20 , ζ 9 20 )	B ≃ E u 20	4
	24	diag(ζ 24 , ζ 5 24 , ζ 7 24 , ζ 11 24 )	B ≃ E u 24	4
	Table 10.3: Classification of possible elements of g in Aut(B, 0), with colored junior
		elements.

Proposition 11.1. Let A be an abelian variety of dimension n, G a group acting freely in codimension 2 on A such that A/G has a crepant resolution X. Then, if g ∈ G is a junior element, the matrix M (g) cannot have eigenvalue 1 with multiplicity exactly n -4.

Remark 11.2. Whether the local affine quotients corresponding to these eight types of junior elements admit a crepant resolution is actually settled by toric geometry in

[START_REF] Sato | Crepant property of Fujiki-Oka resolutions for Gorenstein abelian quotient singularities[END_REF]

. In fact, by

[START_REF] Sato | Crepant property of Fujiki-Oka resolutions for Gorenstein abelian quotient singularities[END_REF] Thm.3

.1], C 4 /⟨iid⟩, C 4 /⟨diag(ω, ω, ω, -1)⟩, C 4 /⟨diag(ζ 8 , ζ 8 , ζ

14.3 Ruling out the pointwise stabilizer

  SL 2 (F 3 ). In this subsection, we prove Proposition 14.2. By Proposition 14.1, it is enough to show the following: Lemma 14.14. Let A be an abelian variety on which a finite group G acts freely in codimension 2 without translations. Suppose that A/G has a simply-connected crepant resolution X. Then there is no abelian subvariety W of codimension 6 in A such that PStab(W ) ≃ SL 2 (F 3 ) < G, with representation M = 1 ⊕n-6 ⊕ σ ⊕3 as in Proposition 14.1.

	This result resembles [5, Sec.6.1], although working under a different set of assump-
	tions and in dimension 6.

16.8. Let

  . By Lemmas 16.5 and 16.6, we have ⟨g, h⟩ ≃ Z 7 . But Z 7 has no proper subgroup, so ⟨g⟩ = ⟨h⟩. Let A be an abelian variety isogenous to E u 7 n . Let g, h ∈ Aut(A) be two junior elements of order 7 such that ⟨g, h⟩ contains no translation and no nonjunior element fixing points. Then E M (g) (1) ∩ E M (h) (1) cannot have codimension 4 in H 0 (T A ). A be an abelian variety isogenous to E j n . Let g, h ∈ Aut(A) be two junior elements of order 3 such that ⟨g, h⟩ contains no translation and no nonjunior element fixing points. Then E

	16.2 Proof of Theorem 16.2. By Theorem 7.6, the proof reduces to the following
	two cases.
	Proposition 16.7. Proposition

  Examples 19.5 and 19.6, we construct explicit examples of fiber products over bases of dimension at least 2, that fail the implication (17.2) ⇒ (17.1). Theorem 17.3 has the following corollary.

	Corollary 17.4. In the setting of Theorem 17.3, assume moreover that for i = 1, 2,
	NE(W

i ) is generated by classes of curves. Let E ∈ Nef(W i ). Then E ∈ Nef(W i ) is extremal, if and only if p * i E ∈ Nef(W ) is extremal.

As a consequence, Nef(W ) is rational polyhedral if and only if both Nef(W 1 ) and Nef(W 2 ) are rational polyhedral.

  Since Ker(ρ) is finite by [23, Corollary 2.11], the first claim follows from [95, Corollary 10.2]. The second statement follows from Theorem 21.8 below.

	Theorem 21.8 ([52, Theorem 1.4]

* (X) is finitely presented.

ACKNOWLEDGMENTS

Acknowledgments. I am grateful to my advisor Andreas Höring for fruitful discussions, and thank Julia Schneider for suggesting Theorem 7.4.

Acknowledgments.

We thank Professor Keiji Oguiso for his suggestions and encouragement. The first author would like to thank JSPS Summer Program for providing the opportunity to visit the third author in Tokyo, where this paper was written. The third author would like to thank Department of Mathematics at National University of Singapore, Professor De-Qi Zhang and Doctor Jia Jia for warm hospitality. The second author is supported by the Ministry of Education Yushan Young Scholar Fellowship (NTU-110VV006) and the National Science and Technology Council (110-2628-M-002-006-). The third author is supported by JSPS KAKENHI Grant Number 21J10242.

Acknowledgments. I am grateful to my advisor A. Höring for regular helpful discussions, to S. Tanimoto for pointing out that the complete intersection of two quadrics in a projective space should satisfy Corollary 22.4, and to J. Cao for suggesting the second question in the introduction.

Step 2: By Step 1 and since ρ is faithful, we know that every element of G W has order 1, 2, 3, or 6. Moreover, there is exactly one element of order 2, namely g 3 , so

Let S be a 3-Sylow subgroup of G W , and s ∈ Z(S) of order 3. Let s ′ ∈ S \ {id A }. By Step 1, every element of ρ(S) other than id B is similar to diag(1, j, j, j), or to diag(1, j 2 , j 2 , j 2 ), in particular, this is the case of s and s ′ , and cannot be both the case of ss ′ and s 2 s ′ , since they commute. Hence, s ′ ∈ ⟨s⟩. So S = ⟨s⟩ ≃ Z 3 , and thus β = 1.

So G W ⊃ ⟨g⟩ has order 6: Hence G W = ⟨g⟩.

Step 3: By [17, Cor.13.2.4, Prop.13.2.5(c)], the number of fixed points of g and g 3 on B are respectively 4 and 256. Let τ be a point of B fixed by g 3 but not by g. By Proposition 10.11, there is a junior element h generating the cyclic group PStab(W + τ ). By Step 2, ⟨h⟩ ⊂ G W = ⟨g⟩. Moreover, as g 3 ∈ PStab(W + τ ) = ⟨h⟩, we know that h has even order, hence order 6 by Proposition 11.3. So ⟨h⟩ = ⟨g⟩, and as both g and h are the only junior elements of order 6 in their generated cyclic groups, g = h. But h fixes τ and g does not, contradiction. Proof of Proposition 11.1. It is straightforward from Propositions 11.3 and 11.5.

PART II FINITE QUOTIENTS OF ABELIAN VARIETIES WITH A CALABI-YAU RESOLUTION

If β = γ = 1, the group F contains an element of order 15, so by Lemma 10.15, B is isomorphic to E u 15 4 and all junior elements of F have order 15. However, since F ≃ Z 15 ⋊ Z 2 α , and since F is generated by its junior elements, we must have α = 0, and so F ≃ Z 15 is cyclic and generated by one junior element.

If β = γ = 0, then F ≃ Z 2 α is cyclic. Else, write p = 3 β 5 γ and F ≃ Z p ⋊ Z 2 α . Note that Z p ⋊ Z 2 α-1 is a proper subgroup of F containing all elements whose order divides 2 α-1 p. As F is generated by its junior elements, their orders cannot all divide 2 α-1 p: There is a junior element g ∈ F of order 2 α or 2 α p. If g has order 2 α p, ⟨g⟩ = F and so F is cyclic. If g has order 2 α , we can write F ≃ ⟨u⟩ ⋊ ⟨g⟩, where u is an element of F of order p. The discussion now depends on α and p.

(1) By Lemma 10.15, if g has order 4, then g = iid commutes with every element of F , so the semidirect product is direct and F is cyclic.

(2) If p = 5 and g has order 8, by Lemma 10.26, g 2 and u commute, so g 2 u has order 20. Since g is junior of order 8, by Lemma 10.15, B is isomorphic to E u 8 4 . So by Lemma 10.15 again, B has no automorphism of order 20, contradiction.

(3) If p = 5 and g has order [START_REF] Birkar | Boundedness of elliptic Calabi-Yau varieties with a rational section[END_REF], by Lemma 10.25, g 4 and u commute, so g 4 u has order 20. But since g is junior of order 16, by Lemma 10.15, B has no automorphism of order 20, contradiction.

(4) If p = 3 and g has order [START_REF] Birkar | Boundedness of elliptic Calabi-Yau varieties with a rational section[END_REF], by Lemma 10.25, g 2 u has order 24. But since g is junior of order 16, by Lemma 10.15, B has no automorphism of order 24, contradiction.

(5) If p = 3 and g has order 8, then F ≃ Z 3 ⋊ Z 8 . With GAP, we check in the Appendix that:

• The irreducible representations of F have rank 1 or 2.

• No irreducible character of F takes value j or j 2 , so F ⊂ Aut(B, 0) has no irreducible subrepresentation of rank 1.

• The only two irreducible representations of F of rank 2 sending -id ∈ F to -id indeed are complex conjugates, so all elements of F ⊂ Aut(B, 0) have characteristic polynomials in Q[X].

However, g ∈ F is a junior element of order 8, which by Lemma 10.15 has a non-rational characteristic polynomial, contradiction.

We prove Proposition 10.27 by contradiction.

Proof of Proposition 10.27. Suppose that 2 divides |F | and that a 2-Sylow subgroup of F is not cyclic. We first show that any junior element in F has order 15, 20 or 24.

By contradiction and by Proposition 9.2, let g ∈ F be a junior element of order 4, 8, 12, or 16. If g has order 12, then g 3 ∈ F is a junior element of order 4, and F thus contains a junior element g of order 4, 8, or [START_REF] Birkar | Boundedness of elliptic Calabi-Yau varieties with a rational section[END_REF]. Let S be a 2-Sylow subgroup containing that junior element. By assumption, S is not cyclic, so by Lemma 10.20, S is isomorphic to Q 8 or to Q 16 . Clearly, Q 8 and Q 16 have no element of order 16, and (4) The abelian subvariety B is in fact G W -stable.

(5) Every h ∈ G W that fixes a point τ ∈ A fixes the point pr B (τ ) ∈ B. [START_REF] Auffarth | Smooth quotients of complex tori by finite groups[END_REF] Moreover, if we assume additionally that there is an integer α ∈ [ [1, d -1]] such that M (g α ) is similar to diag(1 n-k , -1 k ), then, for any h ∈ G W , h and g α commute and

• either there is a point τ ∈ A such that h ∈ PStab(W +τ )∪g α PStab(W +τ );

• or there is no such point, and 1 and -1 are eigenvalues of ρ(h). [START_REF] Auffarth | Smooth quotients of abelian surfaces by finite groups that fix the origin[END_REF] Same assumption. The translation part T (h) of h is a 2-torsion point of B.

(8) Same assumption. If h has even order and fixes a point in A, all fixed points of h in B are of 2-torsion.

(9) Same assumption. If h is a junior element of order 3, then h fixes a 2-torsion point in B.

Proof. (1) follows immediately from [START_REF] Birkenhake | Complex abelian varieties[END_REF]Prop.13.5.1], since M (G W ) is a finite group of group automorphisms of the abelian variety A, and W is M (G W )-stable.

(2) is proven by induction on the number of generators used to write h ∈ G W . First, if h ∈ G W is in G gen , there is a point τ ∈ A such that h ∈ PStab(W + τ ). In particular, for w ∈ W and b ∈ B; M (h)(w + b) = h(w + τ ) -h(τ ) + M (h)(b) = w + ρ(h)(b), as wished. Moreover, T (h) = (id -M (h))(τ ), so pr W (T (h)) = 0.

Second, if h 1 , h 2 ∈ G W satisfy (2), then

since ρ is a group morphism and pr W pr B = pr B pr W = 0. Moreover, T (h 1 h 2 ) = T (h 1 ) + M (h 1 )T (h 2 ), and the fact that pr W (T (h 1 h 2 )) = 0 easily follows from the induction assumption, notably using pr W (id -M (h 1 )) = 0.

For [START_REF] Andreatta | Generalized Mukai conjecture for special Fano varieties[END_REF], let h ∈ G W and note that ρ(h) = id B if and only if M (h) = pr W + pr B = id A , so ρ is faithful since M is. Note that by Proposition 8.4 and Lemma 8.6, M takes values in SL(H 0 (T A )). Hence, by Item 1 of (2), ρ takes values in SL(H 0 (T B )).

Regarding [START_REF] Andreatta | Special rays in the Mori cone of a projective variety[END_REF] we note that, for h ∈ G W , h(B) = M (h)B + T (h) = B + T (h) = B by Item 2 of (2).

(5) is clear from Item 1 of (2).

We now prove [START_REF] Auffarth | Smooth quotients of complex tori by finite groups[END_REF]. Note that ρ(g α ) = -id B commutes with any element of ρ(G W ), and thus, as ρ is faithful, g α is in the center of G W .

Let h ∈ G W and assume that there is no point τ ∈ A fixed by h or g α h. In other words, neither T (h) is in Im(id -M (h)), nor T (g α h) is in Im(id -M (g α h| B )). By Item 2 of (2), T (h) and T (g α h) belongs to B. Hence, the images Im(id B -M (h)| B ) and Im(id B -M (g α h)| B ) = Im(id + M (h)| B ) must be proper subvarieties of B, so 1 and -1 must be eigenvalues of ρ(h) = M (h)| B . 10.11, 11.1 show that if k ≤ 4, then k = 3 and PStab(W ) is cyclic, generated by one junior element. By Proposition 9.2, the junior generator thus has order 3 or 7.

So we can assume k = 5. Up to conjugating the whole group G by a translation, we can assume that 0 ∈ W , and apply Proposition 10.3 to obtain a PStab(W )-stable complementary abelian fivefold B to W . Let F = PStab(W ) ⊂ Aut(B, 0). It is generated by junior elements by Proposition 10.3 (3), which have order 3 or 7 by Propositions 11.3, 11.5, 13.2. Let F ′ be a non-trivial subgroup of F acting not freely in codimension 4: There is an abelian variety W ′ ⊋ W of codimension at most 4 such that F ′ ⊂ PStab(W ′ ). By Propositions 10.11,11.3,11.5, PStab(W ′ ) is cyclic of prime order, so F ′ = PStab(W ′ ) is cyclic generated by one junior element of order 3 or 7.

Note that, by uniqueness of the Poincaré decomposition of B [17, Thm.5.3.7], the group Aut(B, 0) cannot contain both a junior element of order 3 and a junior element of order 7. Hence, if F = PStab(W ) is cyclic, Lemma 10.23 shows that it is generated by one junior element of order p = 3 or p = 7.

To conclude the proof of Proposition 13.1, we thus show by contradiction that F is not cyclic. If F is not cyclic, there are two junior elements g, h ∈ F such that ⟨g, h⟩ is not cyclic, hence acts freely in codimension 4 on B. Let B g and B h be the abelian subvarieties of dimension 3 fixed pointwise by g and h in B. Note that

3 if g and h have order p = 7. Hence, B is accordingly isogenous to E j 5 or to E u 7 5 . So the assumptions of Proposition 13.3 are satisfied, whence F is cyclic, contradiction! To establish Proposition 13.3, we start with a lemma. Lemma 13.4. Let B be an abelian fivefold isogenous to either E j 5 or E u 7 5 , and let p = 3 in the first case, p = 7 in the second case. Let F be a finite subgroup of Aut(B, 0) generated by junior elements of order p, and such that any subgroup of it acting not freely in codimension 4 is cyclic and generated by one junior element of order p. Let g be an element of F of prime order q. Then p = q.

Proof. If 1 is an eigenvalue of g, then ⟨g⟩ acts not freely in codimension 4, so it is cyclic of order p, and p = q.

Suppose that 1 is not an eigenvalue of g. As g has prime order, and by Lemma 2.76, the characteristic polynomial χ g⊕g is a power of the cyclotomic polynomial Φ q . Hence, deg(Φ q ) = q -1 divides 10, so q ∈ {2, 3, 11}. But:

• Since g has determinant 1 and no 1 among its eigenvalues, q ̸ = 2.

Proof of Proposition 13.3. In the notations of Proposition 13.3, Lemma 13.4 proves that F is a p-group. Hence, there is an element g ∈ Z(F ) of order p. Let h ∈ F \ ⟨g⟩ have order p too. Since ⟨g, h⟩ is not cyclic, it must act freely in codimension 4, i.e., E g (1) ∩ E h (1) = {0}, or equivalently the trivial representation is not a subrepresentation of ⟨g, h⟩ ⊂ Aut(B, 0). As g and h commute, they are codiagonalizable. CHAPTER 14

JUNIOR ELEMENTS AND POINTWISE STABILIZERS IN CODIMENSION 6

The goal of this section is to extend the results of Chapters 10, 11, 13 to codimension k = 6. For the first time in our study of pointwise stabilizers, and for the second time in this paper after Section 12, we need to assume the existence of a Calabi-Yau resolution, and not just a crepant (or even simply-connected crepant) resolution of the singular quotient A/G. Indeed, in dimension 6, products of the two examples of [START_REF] Oguiso | On the complete classification of Calabi-Yau threefolds of type III 0[END_REF] yield non-Calabi-Yau simply-connected crepant resolutions of certain singular quotients A/G. We start by proving the following partial classification of pointwise stabilizers in codimension 6 in Subsection 14.1. Proposition 14.1. Let A be an abelian variety on which a finite group G acts freely in codimension 2. Suppose that A/G has a crepant resolution X which is a Calabi-Yau manifold. Let W be a translated abelian subvariety of codimension k ≤ 6 in A such that {1} ̸ = PStab(W ) < G contains no junior element of type diag(1 n-6 , ω, ω, ω, ω, ω, ω). Then PStab(W ) is one of the following.

• A cyclic group generated by one junior element of order 3 or 7.

• An abelian group generated by two junior elements g and h of order both 3 or both 7, satisfying E g (1) ∩ E h (1) = H 0 (W, T W ).

• SL 2 (F 3 ), and the representation M : PStab(W ) → Aut(A, 0) decomposes as

We then use this result to rule out the existence of junior elements with six nontrivial eigenvalues in Subsection 14.2 by a mix of local and global arguments, and finally refine Proposition 14.1 in Subsection 14.3 to the following result. Proposition 14.2. Let A be an abelian variety on which a finite group G acts freely in codimension 2. Suppose that A/G has a crepant resolution X which is a Calabi-Yau manifold. Let W be a translated abelian subvariety of codimension k ≤ 6 in A such that {1} ̸ = PStab(W ) < G. Then PStab(W ) is one of the following.

• A cyclic group generated by one junior element of order 3 or 7.

• An abelian group generated by two junior elements g and h of order both 3 or both 7, satisfying E g (1) ∩ E h (1) = H 0 (W, T W ).

Lemma 14.5. Let B be an abelian sixfold isogenous to either E j 6 or E u 7 6 , and let p = 3 in the first case, p = 7 in the second case. Let F be a finite subgroup of Aut(B, 0) generated by junior elements of order p, such that any subgroup of it acting not freely in codimension 5 is cyclic and generated by one junior element of order p.

Proof. Since -id B is the unique element of order 2 that can belong to F , by [170, 5.3.6], S is cyclic or a generalized quaternion group. Let us show that S has no element of order 8. By contradiction, let s ∈ S be of order 8. Since s 4 = -id B , all eigenvalues of s have order 8, so the characteristic polynomial χ s⊕s is a power of Φ 8 . Comparing degrees yields Φ 8 3 = χ s χ s . But by [START_REF] Washington | Introduction to cyclotomic fields[END_REF]Prop.2.4], Φ 8 is irreducible over Q[j] and

If S is cyclic, then by [170, 10.1.9], there is a normal subgroup N of F such that F = N ⋊ S. But all junior elements of F have odd order, so they belong to N and cannot generate F , contradiction! So S is isomorphic to Q 8 . Lemma 14.6. Let B be an abelian sixfold. Let g ∈ Aut(B, 0) be an element of finite order. Then g cannot have order 27, 49, or 63.

Proof. It is an immediate consequence of Lemma 2.76. Lemma 14.7. Let B be an abelian sixfold isogenous to either E j 6 or E u 7 6 , and let p = 3 in the first case, p = 7 in the second case. Let F be a finite subgroup of Aut(B, 0) generated by junior elements of order p, such that any subgroup of it acting not freely in codimension 5 is cyclic and generated by one junior element of order p. Let q = 7 if p = 3, q = 3 if p = 7. If q divides |F |, a q-Sylow subgroup S of F is cyclic and has order 3, 7, or 9.

Proof. As S is a q-group, there is an element g ∈ Z(S) of order q. Let h ∈ S \ ⟨g⟩ be another element of order q. Because q ̸ ∈ {2, p}, g, h can not be powers of junior elements, and so 1 is not an eigenvalue of them. By Lemma 2.76, g and h are similar to diag(j, j, j, j 2 , j 2 , j

One can then find a on-trivial element of ⟨g, h⟩ with 1 as an eigenvalue. But as g and h commute, it has order q ̸ ∈ {2, p}, contradiction. So ⟨g⟩ is the unique subgroup of order p in S. By [170, 5.3.6], S is thus cyclic, and its order is given by Lemma 14.6.

Lemma 14.8. Let B be an abelian sixfold isogenous to either E j 6 or E u 7 6 , and let p = 3 in the first case, p = 7 in the second case. Let F be a finite subgroup of Aut(B, 0) generated by junior elements of order p, such that any subgroup of it acting not freely in codimension 5 is cyclic and generated by one junior element of order p. Then a p-Sylow subgroup S of F is either cyclic, or the direct product of two cyclic groups. It can be n and G is generated by junior elements of order 7. In particular, G is generated by its elements admitting fixed points. Also note that G contains no junior element of order 6 by Propositions 11.5, 13.2, 14.9.

Let us show that for any translated abelian subvariety W ⊂ A, the pointwise stabilizer PStab(W ) is abelian. It is generated by junior elements by Proposition 10.3. Let g, h be two junior elements in PStab(W ). As g and h both fix abelian varieties of codimension 3, their intersection W ′ has codimension 3, 4, 5, or 6 in A. By Proposition 14.2, PStab(W ′ ) is thus abelian, and therefore g and h commute.

Moreover, any two junior elements g and h in PStab(W ) have the same order

). Hence, using the structure theorem for finite abelian groups, PStab(W ) is isomorphic to

n . Finally, if g, h ∈ PStab(W ) are junior elements, then their eigenspaces with eigenvalues other than 1 are in direct sum by Proposition 14.2. An induction using that all junior elements of PStab(W ) are codiagonalizable then yields Item 3 in Theorem 7.6.

16.1 Proof of Theorem 16.1. By Theorem 7.6, the proof reduces to the following two cases. The first one is simple.

Proposition 16.3. Let A be an abelian variety isogenous to E j

n . Let g, h ∈ Aut(A) be two junior elements of order 3 such that ⟨g, h⟩ contains no translation and no nonjunior element fixing points, and E M (g) (1) = E M (h) (1). Then g = h.

Proof. Recall that M : Aut(A) → Aut(A, 0) which, to any automorphism of A, associates its matrix, induces a representation of ⟨g, h⟩. As ⟨g, h⟩ contains no translation, M is faithful. Applying Maschke's theorem to the invariant subspace E M (g) (1) = E M (h) (1) in H 0 (T A ) yields an ⟨M (g), M (h)⟩-stable supplementary S to it. Let ρ be the faithful representation of ⟨g, h⟩ obtained by restricting M to S. By the classification of junior elements in Proposition 9.2, ρ(g) = ρ(h) = jid B . But ρ is faithful, and thus g = h.

The second case is the following result.

Proposition 16.4. Let A be an abelian variety isogenous to E u 7

n . Let g, h ∈ Aut(A) be two junior elements of order 7 such that ⟨g, h⟩ contains no translation and no nonjunior element fixing points, and (1). Then ⟨g⟩ = ⟨h⟩.

Its proof relies on two lemmas.

Lemma 16.5. Let A be an abelian variety isogenous to E u 7

n . Let g, h ∈ Aut(A) be two junior elements of order 7 such that ⟨g, h⟩ contains no translation and no nonjunior element fixing points, and

Proof. By Maschke's theorem, there is an ⟨M (g), M (h)⟩-stable supplementary S to E M (g) (1) = E M (h) (1) in H 0 (T A ). Consider the faithful representation ρ of ⟨g, h⟩ given by restricting M to S, with character χ.

Let k ∈ ⟨g, h⟩. If k has a fixed point in A, then k is junior of order 7. Else, 1 is an eigenvalue of ρ(k). Since ρ(k) has determinant 1, by Lemma 2.76 and [197,Prop.2.4], the characteristic polynomial of ρ(k) in Q[u 7 ] is one of the following:

So, possible prime divisors of |⟨g, h⟩| belong to {2, 3, 7}. Let S 2 be a 2-Sylow subgroup of ⟨g, h⟩, it inherits the restricted representation ρ| S 2 with character χ| S 2 , and splitting coefficient v 2 . As S 2 has a non-trivial center, A GAP search (see Appendix) through all groups of such order, which have no element of order 12, 14, or 21, and either none or a non-cyclic 2-Sylow subgroup [170, 10.1.9] yields three candidates: Z 7 , Z 7 ⋊ Z 3 , and SL 3 (F 2 ). We exclude the second candidate as it is not generated by its elements of order 7.

We exclude SL 3 (F 2 ) by a geometric argument.

APPENDIX

Groups of order dividing 240 with an automorphism of order 7 [1.. NumberSmallGroups [START_REF] Brozellino | Elementary orbifold differential topology[END_REF]] do g := SmallGroup (24 , v );

if StructureDescription ( g ) = " C3 ␣ : ␣ C8 " then Add ( groups_checked , v ); tbl_conjcl := ConjugacyClasses ( g ); nb_conjcl := Size ( tbl_conjcl ); # locating the unique element of order 2 # among conjugacy classes of g index_2 := 0; for j in [1.. nb_conjcl ] do o := Order ( Representative ( tbl_conjcl [ j ])); if o = 2 then index_2 := j ; fi ; od ; # only keeping irreducible characters sending # the unique element of order 2 to -id T := Irr ( g ); if o = 24 then nb_elts_order_24 := nb_elts_order_24 + s ; fi ; od ; Print ( " number ␣ of ␣ elements ␣ of ␣ order ␣ 24: " ); Print ( nb_elts_order_24 ); Print ( " ␣ " ); Print ( " \ n " ); od ; Print ( " \ n \ n " ); testing := [ [40 ,4] , [40 ,11] , [80 ,18]]; [40 ,11] , [80 ,18]]; Cone Conjecture for Schoen varieties. The main goal of this paper is to apply Theorem 17.3 to a certain type of varieties with globally generated anticanonical bundle, which we call Schoen varieties. This is the higher dimensional generalization of C. Schoen's construction of Calabi-Yau threefolds [START_REF] Schoen | On fiber products of rational elliptic surfaces with section[END_REF] as fiber products over P 1 .

Let us first summarize the construction of Schoen varieties; we refer to Subsections 20.1 and 20.2 for more details. Let Z 1 and Z 2 be Fano manifolds of dimension at least two. For i = 1, 2, let D i be an ample and globally generated divisor on Z i such that -(K Z i + D i ) is globally generated. Let W i ⊂ P 1 × Z i be a general member in the linear system |O P 1 (1) ⊠ O Z i (D i )|. We have a fibration ϕ i : W i → P 1 . Consider the fiber product over P 1 : ϕ :

When X is smooth, such a variety X is called a Schoen variety. It is easy to check that -K X is globally generated, and hence we can define a Schoen pair (X, ∆ m,X ) as in Example 18.1.

We prove the following result.

Theorem 17.5. Let X be a Schoen variety, and let (X, ∆ m,X ) be a Calabi-Yau pair associated to it as in Example 18.1. Then there exists a rational polyhedral fundamental domain for the action of Aut(X, ∆ m,X ) on Nef e (X) = Nef(X).

Note that, by Corollary 17.4, the cone Nef(X) is not rational polyhedral as long as one of Nef(W 1 ) and Nef(W 2 ) is not. This is the case when there exists i such that Z i = P 2 and D i = -K Z i (in which case W i is a rational elliptic surface). In particular, our construction provides the first series of strict Calabi-Yau manifolds, and also Calabi-Yau pairs in arbitrary dimension, for which the Cone Conjecture holds and whose nef cones are not rational polyhedral (see Example 21.6). We also note that X is a complete intersection of two hypersurfaces, which are nef but not ample, in the Fano manifold P 1 × Z 1 × Z 2 . That the cone Nef(X) may admit infinitely many faces resonates with Theorem 17.6 below.

As direct corollaries, we obtain the finite presentation of the discrete group of components π 0 Aut(X) and the finiteness of real structures on X up to equivalence.

Historical remarks. Let us first discuss the state of the art of the Cone Conjecture without the boundary divisor (∆ = 0). The Cone Conjecture was verified for K3 surfaces by Sterk [START_REF] Sterk | Finiteness results for algebraic K3 surfaces[END_REF], and for Enriques surfaces by Namikawa [START_REF] Namikawa | Periods of Enriques surfaces[END_REF] using the Torelli theorem. In [START_REF] Prendergast-Smith | The cone conjecture for abelian varieties[END_REF], Prendergast-Smith proved the Cone Conjecture for abelian varieties. A version of this conjecture was also proven for the two main families of projective hyperkähler manifolds in [START_REF] Markman | A proof of the Kawamata-Morrison cone conjecture for holomorphic symplectic varieties of K3 [n] or generalized Kummer deformation type[END_REF], shortly before the general proof by Amerik-Verbitsky came out in [START_REF] Amerik | Morrison-Kawamata cone conjecture for hyperkähler manifolds[END_REF].

Very little is known about the Cone Conjecture for strict Calabi-Yau manifolds (see Definition 18.2), even in dimension three. The most general result might be the following, due to Kollár [START_REF] Borcea | Homogeneous vector bundles and families of Calabi-Yau threefolds. II. In Several Complex Variables and Complex Geometry[END_REF].

Theorem 17.6. Let D be a smooth anticanonical hypersurface in a Fano manifold Y of dimension at least 4. Then the natural restriction map Nef(D) → Nef(Y ) is an isomorphism. In particular, Nef(D) is a rational polyhedral cone.

Among the strict Calabi-Yau manifolds whose nef cones are not rational polyhedral, to our knowledge the Cone Conjecture is known so far for only two special cases.

There is a rational polyhedral fundamental domain for the action of ρ(H) on

Nef + (X).

The group ρ(H) is finitely presented.

Such a result and related statements are well-known to experts. We include a proof for the sake of completeness. It relies on the fundamental results due to Looijenga [132, Application 4.14 and Corollary 4.15], which we extract and formulate here as Lemma 18.4. Recall that a cone C ⊂ N R in a finite dimensional R-vector space N R is called strict if its closure C ⊂ N R contains no line.

Lemma 18.4. Let N be a finitely generated free Z-module, and let C be a strict convex open cone in the R-vector space

R be the interior of the dual cone of C. Let Γ be a subgroup of GL(N ) which preserves the cone C. Suppose that:

• there is a rational polyhedral cone

Then the Γ-action on C + has a rational polyhedral fundamental domain, and the group Γ is finitely presented.

Proof of Proposition 18.3. In Lemma 18.4, now set N = N 1 (X), C = Amp(X), and Γ = ρ(H). Let us construct an element ξ ∈ (C ∨ ) • ∩ N ∨ Q whose stabilizer is trivial. If we have such an element, then by Lemma 18.4, Proposition 18.3 follows.

Claim 18.5. There exists an ample class η

Proof. Our proof is inspired by the argument of [START_REF] Lazić | Around and beyond the canonical class[END_REF]Proposition 6.5].

By Fujiki-Liebermann's theorem [23, Theorem 2.10], the action of Γ on C ∩ N has finite stabilizers. Take an element η ∈ C ∩ N Q such that the order of the stabilizer Γ η is minimal. Since the Γ-action on N R preserves N , we can find an open neighborhood U ⊂ C of η, such that γU ∩ U = ∅ for every γ / ∈ Γ η . Thus, for every η ′ ∈ U ∩ N Q , we have Γ η ′ ⊂ Γ η , which then implies Γ η ′ = Γ η by the minimality of Γ η . It follows that every γ ∈ Γ η satisfies γ| U ∩N Q = id U ∩N Q , and since γ acts linearly, necessarily γ = id. This proves that η ∈ C ∩ N Q has trivial stabilizer, and so do some positive multiple

there are only finitely many η ∈ Σ minimizing ξ| Σ . Again, as C ∩ N is discrete, we can perturb ξ and obtain

for every γ / ∈ Γ η . Since η ∈ Σ, the stabilizer Γ η is trivial, so the stabilizer of ξ 0 in Γ is trivial as well.

CHAPTER 19

THE NEF CONE OF A FIBER PRODUCT OVER A CURVE

We now prove Theorem 17.3 about the decomposition of the nef cone.

For i = 1, 2, recall that ϕ i : W i → B is a surjective morphism with connected fibers from a smooth projective variety to a smooth projective curve B. We consider the fiber product

and work under the following assumptions:

Proof of Theorem 17.3. Let D ∈ Nef(W ) and let

be a decomposition as in [START_REF] Amerik | Morrison-Kawamata cone conjecture for hyperkähler manifolds[END_REF]. First, note the following simple fact.

Proof. We may only consider i = 1. Choose any point s ∈ ϕ -1 2 (ϕ 1 (C 1 )) and let

This proves the assertion.

We use this fact to prove the following two lemmas.

Proof. We may only consider the case when i = 2. Let C 1 ⊂ W 1 be an irreducible curve such that ϕ 1 (C 1 ) = B. Define

.

By construction, we have

This shows that D ′ 2 is nef. Hence, for n ≥ N 2 , the divisor

We can now resume the proof of Theorem 17.3. For any t ∈ R, let

. By Lemma 19.3, there exist Example 19.5. Take S := P 2 , and take four points P 1 , P 2 , P 3 , P 4 in S so that no three of them lie on a line. Let ℓ 1 be the line through P 1 , P 2 , and let ℓ 2 be the line through P 3 , P 4 . Take W 1 := Bl P 1 ,P 2 (S) and W 2 := Bl P 3 ,P 4 (S).

As the blown-up points are distinct, W := W 1 × S W 2 is isomorphic to Bl P 1 ,P 2 ,P 3 ,P 4 (S), which is smooth. Moreover, the decomposition of the Picard group

clearly holds. Denote by ℓ ′ 1 and ℓ ′ 2 the strict transforms of ℓ 1 and ℓ 2 in W 1 and W 2 respectively. Then ℓ ′ i is an effective non-nef divisor on

We show that D is nef; this also shows that Lemma 19.2 fails when dim B ≥ 2. As D is effective, it is enough to check that its intersections with its components are all non-negative. By symmetry, it is enough to compute

So D is nef, and has vanishing intersection with the curves p * 1 ℓ ′ 1 and p * 2 ℓ ′ 2 .

Now assume by contradiction that D has another decomposition

where p : W → S is the blow up, and ℓ is a line passing through none of P 1 , P 2 , P 3 , P 4 in S. It follows that

As for examples of fiber products over a base of higher dimension, we continue with the notations of Example 19.5, and introduce

where T is an arbitrary smooth projective variety. As in Example 19.5, W, W 1 and W 2 are rationally connected, hence have trivial irregularity, so that

Note that by the projection formula,

We have the following corollary of Theorem 17.3.

Corollary 19.7. For i = 1, 2, let ϕ i : W i → B be a surjective morphism with connected fibers from a smooth projective variety to a smooth projective curve B. Assume that CHAPTER 20

CONSTRUCTION OF SCHOEN VARIETIES

Schoen varieties will be constructed as a fiber product of two fibrations over P 1 . Let us first construct these fibrations.

20.1

The factor W with a fibration over P 1 . The construction relies on a pencil of ample hypersurfaces in a Fano manifold.

Let Z be a Fano manifold of dimension at least 2, and let D be an ample divisor in Z such that both O Z (D) and O Z (-K Z -D) are globally generated. Note that O Z (-K Z ) is then globally generated as well.

Example 20.1. Take any toric Fano manifold Z. Since nef line bundles on a projective toric manifold are globally generated, any decomposition -K Z = D + D ′ as the sum of an ample divisor D and a nef divisor D ′ yields a pair (Z, D) satisfying the above condition.

Let W ⊂ P 1 ×Z be a general member of the ample and basepoint-free linear system |O P 1 (1) ⊠ O Z (D)|. We have a fibration ϕ : W → P 1 via the first projection, and the second projection ε : W → Z is the blow-up of Z along the smooth subvariety Y of codimension two cut out by the members of the pencil in |D| defined by W . Since Z is Fano, W is rationally connected. By construction, the rational curve ε -1 (y) ≃ P 1 for any y ∈ Y is a section of ϕ :

by the adjunction formula. So O W (-K W ) is globally generated, in particular, nef and effective.

The following lemma describes the possibilities for W in dimension 2. Recall that a smooth projective surface S is called weak del Pezzo if its anticanonical divisor -K S is nef and big. Proof. Since W is rationally connected and dim W = 2, we know that W is rational.

. So -K W is globally generated and W is a rational elliptic surface.

Suppose that D / ∈ | -K Z |. As -K Z -D is effective and -K Z and D are ample, we have -K Z (-K Z -D) > 0 and D(-K Z -D) > 0, and thus,

PART IV

POSITIVITY OF HIGHER EXTERIOR POWERS OF THE TANGENT BUNDLE

CHAPTER 23

A FIRST LEMMA

We start with a simple lemma.

Lemma 23.1. Let X be a smooth projective variety of dimension n, and let 1 ≤ r ≤ n -1. The following results hold:

Proof. The proof goes as [128, Proof of Theorem 1.5]. Let f : P 1 → C be the normalization of the curve. Write

with (a i ) 1≤i≤n ordered increasingly. It holds a n ≥ 2, as T P 1 maps non-trivially to f * T X , and we have a 1 + . . . + a r ≥ 0 because O P 1 (a 1 + . . . + a r ) is a direct summand of the nef vector bundle r f * T X . Moreover, if r T X is strictly nef, the inequality is strict, and in particular, a r+1 ≥ a r ≥ 1. Hence, if r T X is strictly nef, we obtain

This result is all the more valuable as, by [128, Theorem 1.2], if X is a smooth projective variety of dimension n such that r T X is strictly nef, then it is rationally connected, in particular, it contains numerous rational curves. CHAPTER

25

STUDYING MORI CONTRACTIONS

The strategy for proving Theorems 22.3 and 22.5 is to show that there are only few possible birational contractions for X. In the following, if R is an extremal ray of the Mori cone N E(X), its length denoted by ℓ(R) is defined to be the minimal value of -K X • C, for a rational curve C with class in R. A Mori contraction is said to be of length ℓ if it is a contraction of a ray R with ℓ(R) = ℓ.

Small contractions.

Lemma 25.1. Let r ∈ [ [1,[START_REF] Andreatta | Special rays in the Mori cone of a projective variety[END_REF]]. Let X be a smooth projective variety of dimension at least r + 1 such that r T X is strictly nef. Then X has no small contraction.

Proof. Let n be the dimension of X. Let φ : X → Y be a birational contraction, E be an irreducible component of the exceptional locus, F an irreducible component of the general fiber of φ| E , and R the corresponding extremal ray. Applying Ionescu-Wiśnewski inequality [92, Theorem 0.4], [206, Theorem 1.1] together with Lemma 23.1

Since r ≤ 4, we have dim E ≥ n -1, and thus φ is a divisorial contraction.

Fibred Mori contractions.

We move on to studying fibred Mori contractions.

Generalities about fibred Mori contractions

We use families of deformations of rational curves (see Section 2.12) to prove the following proposition.

Proposition 25.2. Let X be a smooth projective rationally connected variety of di-

Then the general fiber of π has dimension at most r -1.

If equality holds, then there is a rational curve C in X, not contracted by π, whose family of deformations V is unsplit covering and satisfies dim Locus(V x ) = n + 1 -r for x ∈ Locus(V) general.

Proof of Proposition 25.2. Since X is rationally connected and -K X is Cartier, we dispose of a rational curve C such that π(C) ̸ = {pt} and -K X • C ≥ n + 2 -r ≥ 3 is minimal with this condition. Let V be the corresponding family of deformations. By Lemma 2.123, it is unsplit.

Fix x ∈ Locus(V) general. By [108, Proposition IV.2.6] and Lemma 23.1, we derive

Let d denote the dimension of the general fiber of π. Then, by Lemma 2.124,

As for the equality case, if d = r -1, then dim Locus(V x ) = n -r + 1, and so C is such a rational curve as we claimed existed in the equality case of the proposition. Proposition 25.2 has an important consequence.

Corollary 25.3. Let X be a smooth projective rationally connected variety of dimension n such that, for some r ∈

If equality holds, then a general fiber of π has dimension r -1, and there is a rational curve C in X, not contracted by π, whose family of deformations V is unsplit covering and satisfies dim Locus(V x ) = n + 1 -r for x ∈ Locus(V) general.

Proof. Let F be a general fiber of π. By Proposition 25.2, we have r -1 ≥ dim F . Adding n to both sides and applying Ionescu-Wiśnewski inequality (with the exceptional locus E = X of dimension n), it holds

If there is an equality, then dim F = r -1, and so we are in the equality case of Proposition 25.2.

Fibred Mori contractions for certain varieties of even dimension

The set-up for this paragraph is the following. Let r be 3 or 4. Let X be a smooth projective rationally connected variety of dimension 2r -2 such that -K X • C ≥ r for any rational curve C ⊂ X. Suppose that there is a fibred contraction π : X → Y with dim Y > 0. Let us classify what happens.

Lemma 25.4. Let r be 3 or 4. Let X be a smooth projective rationally connected variety of dimension 2r -2 such that -K X • C ≥ r for any rational curve C ⊂ X. Suppose that there is a fibred contraction π : X → Y with dim Y > 0. Then there is another equidimensional fibred Mori contraction φ : X → Z with dim Z = r -1.

Proof. We are in the case of equality of Corollary 25.3. In particular, the general fiber F of π has dimension r -1, and there is a rational curve C in X that is not contracted by π whose family of deformations V is unsplit covering and satisfies dim Locus(

By [18, Theorem 2, Proposition 1(i)], there is a fibred Mori contraction φ : X → Z whose fibers exactly are the V-equivalence classes, and its general fiber has dimension dim Locus(V x ) = r -1.

Let G be a fiber of φ. We claim that π| G is finite. Indeed, if it is not, then there is a curve B ⊂ G that is contracted by π. The curve B lies in a V-equivalence class, so by [18, Remark 1], as V is unsplit, B is numerically equivalent to a multiple of C, so it cannot be contracted by π, contradiction! So π| G is finite onto its image, which is contained in Y , so dim G ≤ dim Y = r -1.

So φ is indeed equidimensional.

Proposition 25.5. Let r ≥ 3 be an integer. Let X be a smooth projective rationally connected variety of dimension 2r -2 such that -K X • C ≥ r for any rational curve C ⊂ X. Suppose that there is an equidimensional fibred Mori contraction π :

This proposition relies on the following lemma.

Definition 25.6. Let π : X → Y be a fibration whose general fiber is a projective space. Let f : P 1 → C ⊂ Y be a rational curve whose image lies in the smooth locus of π. The fiber product π C of π by f is the projectivization of a bundle O P 1 (a 1 ) ⊕ . . . ⊕ O P 1 (a k ), with the (a i ) ordered increasingly. A minimal section over C is the section s : P 1 → X of π C corresponding to a quotient O P 1 (a 1 ).

Remark 25.7.

There may be several minimal sections as soon as a 1 = a 2 .

Lemma 25.8. Let X be a smooth projective variety with a fibration π : X → Y whose general fiber is a projective space. Then for any rational curve f : P 1 → C ⊂ Y 0 ⊂ Y in the smooth locus of π, for any minimal section s of it, it holds -K Y • C ≥ -K X • s(P 1 ).

In particular,

If there is an equality in ( * * ), then the base change of π by f is isomorphic to

If there is almost an equality, i.e.,

then the base change of π by f is isomorphic to to P(O P 1 ⊕k ) → P 1 or to P(O P 1 ⊕k-1 ⊕ O P 1 (1)) → P 1 .

Proof. By Tsen's theorem, the base change π C of π by f is the natural projection morphism of a projectivized vector bundle V on P 1 . We write V ≃ O P 1 (a 1 ) ⊕ . . . ⊕ O P 1 (a k ), with (a i ) ordered increasingly, and consider s the section of π C satisfying s * O P(V ) (1) = O P 1 (a 1 ). The degree of det(s * O P(V ) (1))⊗V * is non-positive, equals zero if and only if V ≃ O P 1 (a 1 ) ⊕k , and equals one if and only if V ≃ O P 1 (a 1 ) ⊕k-1 ⊕O P 1 (a 1 +1).

Pulling-back the Euler exact sequence of π C by s, we get

Thus, s * T X/Y has non-positive degree. We also have the tangent bundle exact sequence: 0

Proof of Lemma 25.10. Since X is smooth and X → Y is a projective bundle, Y is smooth as well. Since X is rationally connected, Y is rationally connected and by Lemma 25.8, one has -K Y • C ≥ 3 for any rational curve C in Y . By [START_REF] Dedieu | Numerical characterisation of quadrics[END_REF]Cor.1.4], Y is a quadric hypersurface Q 3 or the projective space P 3 . In either case, Y is rational and so it has trivial Brauer group. Hence, X = P(V ) for some vector bundle V on Y .

If Y is a quadric, then all lines ∆ ⊂ Y satisfy -K Y • ∆ = 3, and thus by the equality case in Lemma 25.8, V | ∆ ≃ L ∆ ⊕3 for some line bundle L on ∆. Fixing a line ∆ 0 , we have, as

for any line ∆ in Y . By Lemma 25.11, this twist of V is globally trivial and thus X ≃ P 2 × Q 3 .

Suppose now that Y is a projective space. By the almost-equality case in Lemma 25.8, for every line ∆ in Y ,

If it is divisible by 3, then we are in the first case, else it is congruent to 1 modulo 3 and we are in the second case. In both cases, the a i,∆ are thus independent of the line ∆. Fixing a line ∆ 0 , the restricted twisted bundle (V ⊗ O P 1 (-a 1,∆ 0 ))| ∆ therefore is a uniform bundle of type (0, 0, 0) or (0, 0, 1). In the first case, this twist of V is globally trivial by [START_REF] Okonek | Vector Bundles on Complex Projective Spaces[END_REF], and so X ≃ P 2 × P 3 . In the second case, by [START_REF] Sato | Uniform vector bundles on a projective space[END_REF], this twist of V is either O P 3 ⊕ O P 3 ⊕ O P 3 (1) or T P 3 (-1), which concludes the classification.

Let us now study a more general fibred Mori contraction of X. Lemma 25.12. Let X be a smooth projective rationally connected fivefold and such that, for any rational curve C ⊂ X, one has -K X • C ≥ 3. Suppose that X has a fibred Mori contraction π : X → Y . Then dim Y ≤ 3.

Proof. If dim(Y ) = 4, the general fiber of π is a smooth curve C with trivial normal bundle. By assumption,

Let us cover the case when X has a fibred Mori contraction π : X → Y with 1 ≤ dim(Y ) ≤ 2. Lemma 25.13. Let X be a smooth projective rationally connected fivefold and such that, for any rational curve C ⊂ X, one has -K X • C ≥ 3. Suppose that X has a fibred Mori contraction π : X → Y with 1 ≤ dim Y ≤ 2. Then there is a fibred Mori contraction p : X → Z that is a P 2 -bundle.

Proof. We dispose of a rational curve C such that π(C) ̸ = {pt} and -K X • C ≥ 3 is minimal with this condition. Let V be the corresponding family of deformations. By Proof. Let ε : X ′ → X be a resolution of indeterminacies for f , let f ′ : X ′ → Y be the induced holomorphic map. As f is almost holomorphic, no component of the exceptional locus of ε is dominant onto Y . As Y is curve, this means that the exceptional locus of ε is sent onto finitely many points in Y . So f ′ factors through ε, i.e., f is holomorphic. Lemma 25.18. Let X be a smooth projective rationally connected fivefold, such that -K X • C ≥ 3 for any rational curve C ⊂ X. Suppose that X has a fibred Mori contraction π : X → Y with dim Y = 3. If π is not a P 2 -bundle, then any rational curve C ⊂ X such that π(C) ̸ = {pt}, and which deforms in an unsplit family, deforms in a family covering X.

Proof. Note that if π is equidimensional, by [START_REF] Höring | Mori contractions of maximal length[END_REF]Theorem 1.3] it is a P 2 -bundle. Hence, we assume that a variety F of dimension 3 is contained in a fiber of π. By contradiction, we consider a rational curve C ⊂ X such that π(C) ̸ = {pt}, and the family V of deformations of C is unsplit and not covering X.

Fix x ∈ Locus(V) general. By Lemma 2.124, dim Locus(

in particular as V is not covering, dim Locus(V) = 4 and dim Locus(V x ) = 3.

Let n : D → D denote the normalization of D = Locus(V), and let Ṽ be the covering family on D. Note that π induces a fibration of D onto a variety of smaller dimension that is not a point, in particular ρ( D) ≥ 2. Thus, by [START_REF] Andreatta | Generalized Mukai conjecture for special Fano varieties[END_REF]Corollary 4.4], D cannot be Ṽ-chain-connected.

Considering the dominant almost holomorphic map r : D Z whose general fiber is a Ṽ-equivalence class [18, Section 2], the variety Z is thus not a point. Since dim Locus( Ṽx ) = 3 for a general x ∈ Locus( Ṽ), the variety Z must be a curve, in particular, by Lemma 25.17, the map r is holomorphic.

Note that, as D is a relatively ample Cartier divisor with respect to π, it intersects the three-dimensional variety F along a surface S. Since dim n -1 (S) = 2 > dim Z = 1, the restriction r| n -1 (S) : n -1 (S) → Z cannot be finite. So it contracts a curve B. Its image n(B) is in a V-equivalence class, so as V is unsplit, it is numerically equivalent to a multiple of C. But n(B) ⊂ F , so this curve is contracted by π, contradiction. Definition 25.19. Let f : X → Y be a finite surjective map. We say that f is quasiétale if it is étale in codimension 1.

Remark 25.20. Note that if f : X → Y is quasiétale and Y is smooth, then by Zariski purity of the branch locus, f is étale. Lemma 25.21. Let X be a smooth projective rationally connected fivefold, such that -K X • C ≥ 3 for any rational curve C ⊂ X. Suppose that X has a fibred Mori contraction π : X → Y with dim Y > 0. If X is not a P 2 -bundle over any smooth projective base, then Y ≃ P 3 . Moreover, ρ(X) = 2, and if C is a line in the smooth locus Y 0 ⊂ Y of π and s a minimal section over C in X, the class of s(P 1 ) generates the other extremal ray in N E(X), induces a fibred Mori contraction to a positive dimensional variety too, and satisfies -K X • s(P 1 ) = 3. Proof of Theorem 22.3. Note that -K X is nef, and non-trivial (as it is positive on rational curves, and X is rationally connected) As T P n | X ⊗ O X (-1) is nef, the quotient of its fourth exterior power 3 T X ⊗ O X (-1) is also nef, and thus 3 T X is ample. As T P n | X ⊗ O X (-1) is nef, the quotient of its fifth exterior power 3 T X ⊗ O X (-1) is also nef, and thus 3 T X is ample.

Examples for Corollary 22.6.

Lemma 26.4. Let X be the fivefold P(T P 3 ). Then 4 T X is ample.

Proof. Denote the natural projection by p : X → P 3 , the tautological line bundle on X by O X (1). By [START_REF] Hartshorne | Algebraic Geometry[END_REF]II.Ex.5.16(d)], there is an exact sequence 0 → Let us prove that E 1 = T X/P 3 ⊗ p * O P 3 (-K P 3 ) is ample. We have the relative Euler sequence 0 → O X → p * Ω 1 P 3 ⊗ O X (1) → T X/P 3 → 0. The bundle E 1 is a quotient of p * Ω 1 P 3 (4) ⊗ O X (1). But as T P 3 is ample, O X (1) is ample. Moreover, Ω 1 P 3 (4) ≃ 2 T P 3 is ample too, which concludes by [122, 6.1.16]. Let us prove that E 2 = 2 T X/P 3 ⊗ p * 2 T P 3 is ample. This would settle the ampleness of 4 T X by [122, 6.1.13(ii)]. From [81, II.Ex.5.16(d)] and the relative Euler sequence, we derive 0 → T X/P 3 → p * T P 3 (-4) ⊗ O X (2) → 2 T X/P 3 → 0.

Since E 2 is a quotient of p * (T P 3 (-4) ⊗ 2 T P 3 ) ⊗ O X (2), we are left proving that the latter is ample. Notice that T P 3 (-1) is globally generated and thus nef. So the bundle T P 3 (-3) ⊗ 2 T P 3 = T P 3 (-1) ⊗ 2 T P 3 (-1) is nef as well. Finally, O X (1) is ample, and we see that O X (1) ⊗ p * O P 3 (-1) is a quotient of p * T P 3 (-1) (dualizing the relative Euler exact sequence and twisting by O X (1)), hence it is nef. We conclude by [122, 6.2.12(iv)]. Proof. Denote the natural projection by p : X → P 3 , the tautological line bundle on X by O X (1). By [START_REF] Hartshorne | Algebraic Geometry[END_REF]II.Ex.5.16(d)], there is an exact sequence 0 → It is thus enough to prove that p * 2 T P 3 ⊗ p * O P 3 (-1) ⊗ O X (2) is ample, which is clear since 2 T P 3 (-1) = ( 2 T P 3 )(-2) is globally generated and thus nef, and since p * O P 3 (1) ⊗ O X (2) is ample. Remark 26.7. It is easy check to that 4 T P 2 ×P 3 , 4 T P 2 ×Q 3 , 4 T P 3 ×P 3 are ample. Proof of Theorem 22.5. Note that -K X is nef, and non-trivial (as it is positive on rational curves, and X is rationally connected). If ρ(X) = 1, -K X is ample and X is thus a Fano variety. If ρ(X) ≥ 2, by the Cone Theorem, X admits a Mori contraction. By Lemma 25.1, it cannot be a small contraction.

Suppose that it is a divisorial contraction. By Corollary 25.28 and Lemma 25.29, it is a smooth blow-up of a smooth curve in a fivefold, but we are assuming that X has dimension at least six, contradiction! So X has no divisorial contraction. Thus, it has a fibred Mori contraction onto a positive dimensional variety. Corollary 25.3 implies that X is a fivefold or a sixfold. By assumption, X is thus a sixfold. By Lemma 25.4, X has an equidimensional fibred Mori contraction to a threefold, so by Proposition 25.5, we have X ≃ P 3 × P 3 , which concludes.

Proof of Corollary 22.6. By Theorem 22.5, is is enough to consider the case when X is a fivefold. In particular, by Theorem 22.2, X is a Fano variety. Again, if ρ(X) = 1, there is nothing to prove.

If ρ(X) ≥ 2, by the Cone Theorem, X admits a Mori contraction. By Lemma 25.1, it cannot be a small contraction.

Suppose that it is a divisorial contraction. By Corollary 25.28, it is a smooth blow-up of a smooth curve, and by Proposition 25.31, X ≃ Bl ℓ P 5 .

Otherwise, it is a fibred Mori contraction onto a positive dimensional variety. Since X is a fivefold such that 4 T X is strictly nef, Proposition 25.9 applies and concludes.