Î T Beno

Stefan Claudon

AND Behrouz Kebekus
email: behrouz.taji@math.uni-freiburg.de

Taji

Benoît Claudon
email: benoit.claudon@univ-lorraine.fr

Institut Elie

Cartan Nancy

Kebekus

GENERIC POSITIVITY AND APPLICATIONS TO HYPERBOLICITY OF MODULI SPACES

Keywords: 28th Mathematics Subject Classification. 14D23, 14E05, 14E30, 14F10, 14J10, 14J17, 32Q45, 32Q26 families, canonically-polarized manifolds, moduli, Kodaira dimension, foliation, generic semipositivity, minimal model program

The proof of the celebrated Viehweg's hyperbolicity conjecture is a consequence of two remarkable results: Viehweg and Zuo's existence results for global pluri-differential forms induced by variation in a family of canonically polarised manifolds and Campana and Pȃun's vast generalisation of Miyaoka's generic semipositivity result for non-uniruled varieties to the context of pairs. The aim of this chapter is an exposition of Campana-Pȃun's generic semipositivity theorem.

INTRODUCTION

In 1962 Shafarevich conjectured that a smooth family f • : X • → Y • of complex projective curves of genus at least equal to 2, parameterized by Y • = P 1 , C, C * , or an elliptic curve E is isotrivial, so that is there is no variation in the algebraic structure of the members of the family. Equivalently this conjecture can be expressed as the prediction that the base Y • of any smooth, non-isotrivial family of projective curves with g ≥ 2 is of log-general type. In other words, we have κ(Y, K Y + D) = 1 for any smooth compactification (Y, D) of Y • , with snc boundary divisor D. Shafarevich conjecture was shown by Parshin and Arakelov.

To generalise the Shafarevich conjecture to higher dimensional fibres and parametrizing spaces, Viehweg considered the hyperbolicity properties of the moduli stack of canonically-polarised manifolds. Recall that the moduli functor M of canonically-polarised manifolds with fixed Hilbert polynomial, is equipped with a natural transformation

Ψ : M(•) → Hom(•, M),
where M denotes the coarse moduli scheme associated with M. The scheme M was proved by Viehweg to be quasi-projective, cf. [START_REF] Viehweg | Quasi-projective moduli for polarized manifolds[END_REF]. Also recall that a complex analytic space U is said to be Brody hyperbolic if there are no non-constant holomorphic maps f : C → U. In the spirit of this definition, Shafarevich's conjecture is equivalent to the assertion that the base Y • of non-isotrivial, smooth, projective families of high genus curves is algebraically Brody hyperbolic in the sense that there are no non-constant morphisms from C * to Y • . Generalising Shafarevich's conjecture, Viehweg predicted that the moduli stack of canonically-polarised manifolds is not only algebraically Brody hyperbolic but that it is Brody hyperbolic. More precisely, a smooth quasi-projective variety Y • admitting a generically finite morphism µ : Y • → M, must be Brody hyperbolic. This conjecture was settled by Viehweg and Zuo in [START_REF] Viehweg | On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds[END_REF]. On the other hand, a long-standing conjecture of Lang predicts that for a quasi-projective Y • , Kobayashi hyperbolicity (which is equivalent to Brody hyperbolicity for projective varieties) implies that all subvarieties of Y • , including Y • , are of log-general type. In the light of Lang's problem, Viehweg extended his question on the hyperbolic nature of the moduli stack of canonically-polarised manifolds to the following conjecture.

Conjecture 1.1 (Viehweg's hyperbolicity conjecture). Let Y • be a smooth quasiprojective variety admitting a generically finite morphism µ : Y • → M. Then, the smooth compactification (Y, D) of Y • is of log-general type.

Viehweg's conjecture has attracted the interest of many algebraic geometers for a long time. We refer the reader to the survey [Keb13] for more details, including references to earlier results that are not mentioned here for lack of space.

1.1. Viehweg's hyperbolicity conjecture according to Viehweg-Zuo and Campana-Pȃun. A general strategy to prove Conjecture 1.1 consists of two main steps. Combining deep results of analytic [Zuo00], algebraic [Vie83] and Hodge theoretic [Gri84] nature, Viehweg and Zuo construct in a first step a subsheaf of the sheaf of pluri-log differential forms of the base whose birational positivity captures the variation 1 in the family.

Theorem 1.2 (Existence of pluri-logarithmic forms in the base, cf. [VZ02, Thm. 1.4]). If the smooth family of canonically-polarised manifolds f • has maximal variation, then there exist a positive integer N an invertible subsheaf L ⊆ Sym N Ω 1 Y log(D) such that κ(Y, L ) = dim Y. Theorem 1.2 immediately resolves the original conjecture of Shafarevich. The goal in the second step is to trace a connection between the birational positivity (bigness) of L in Theorem 1.2 and that of K Y + D, thus resolving Conjecture 1.1. Working along these lines, the second author and Kovács established Conjecture 1.1 for moduli stacks of dimension two and three, [KK08, KK10] and see [Keb13] for an overview. The work relied, among other things, on the logabundance theorem for surfaces and threefolds. In the absence of these methods in higher dimensions, for instance a complete solution to the abundance problem, Campana and Pȃun devised an additional tool, namely a vast generalisation of the famous generic semipositivity result of Miyaoka to the context of pairs with rational coefficients. Here, we state their result in its simplest form and we refer the reader to Section 5 for a general statement.

Theorem 1.3 (Logarithmic generic semipositivity, cf. [CP15b, Thm. 2.1]). Let (X, D) be a reduced, projective, snc pair. If K X + D is pseudo-effective, then for every ample divisor H on X and every torsion free quotient Q of Ω 1 X (log D) we have

c 1 (Q) • [H] n-1 ≥ 0.
Despite its importance, we found the paper [CP15b] rather hard to read. This chapter is meant to serve as an exposition of Campana and Pȃun's proof of Theorem 1.3 and its application to resolving Conjecture 1.1. 1.2. Structure of the current chapter. In Section 2 we gather some preliminary definitions and notions that are used throughout this chapter. In Section 3 we review some of the basics of the theory of orbifolds. In Section 4 we delve deeper into some technical details that will be crucial to the proof of the generic semipositivity result in Section 8. In Section 5 we state Theorem 1.3 in its full generality. Section 6 sketches the proof of Conjecture 1.1 using this result. Part II is devoted to the proof of the semipositivity result of Campana and Pȃun. More recently, they strengthened Theorems 1.3 and 5.3 also in another direction, by proving the pseudo-effectivity of torsion free quotients, [CP15a]. This latter result is specially significant for the proof of Viehweg's conjecture, as it makes the LMMP methods redundant. For a concise exposition of [CP15a] and its application to Viehweg's problem we refer the reader to the first author's notes written for the Bourbaki seminar, [Cla15].

In a slightly different, but closely related, direction a more general version of Viehweg's conjecture, that is perhaps closer to the spirit of the original conjecture of Shafarevich, was formulated by Campana. In this conjecture Campana proposed the so-called special varieties as higher dimensional analogues of C, C * , P 1 and E in Shafarevich conjecture. We refer the reader to the original paper of Campana, [Cam04], for the basic definitions and background in the theory of special varieties.

Conjecture 1.4 (The isotriviality conjecture). Any smooth family of canonicallypolarised manifolds f

• : X • → Y • parametrised by a special quasi-projective variety Y • is isotrivial.
Following the strategy of Campana and Pȃun and by using the result of [JK11b], Conjecture 1.4 has been settled in [START_REF] Taji | The isotriviality of smooth families of canonically polarized manifolds over a special quasi-projective base[END_REF]. More recently, in [PS15], Popa and Schnell have proved a vast generalisation of Conjecture 1.1 by extending Theorem 1.2 to smooth projective families of varieties of general type. Where their strategy follows the same two-steps approach discussed above, the main breakthrough in their result comes from an interesting use of the theory of Hodge modules to extend some crucial Hodge theoretic tools used in [START_REF] Viehweg | On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds[END_REF].

1.4. Acknowledgements. The authors owe a special thanks to Frédéric Campana and Mihai Pȃun for many fruitful discussions.

DEFINITIONS AND NOTATION

In the current section we gather some very basic definitions and concepts needed for the arguments in the later parts of this chapter. For the more standard definitions, we refer to [Har77]. The reader who is familiar with these preliminaries may wish to skip Subsections 2.1 to 2.5 and move on Subsection 2.6. In this chapter, all varieties are defined over C.

2.1. Varieties, subsets, sheaves and pairs. Let us begin by introducing the most basic objects, recurrent throughout this chapter.

Notation 2.1 (Small and big sets). Let X be a variety. A subset S ⊆ X is called small if its Zariski closure satisfies codim X S ≥ 2. A subset U ⊆ X is called big if its complement is small. Notation 2.2 (Families of curves on projective varieties). Let X be a projective variety. A family of curves is a smooth subvariety T ⊆ Hilb(X) whose associated subschemes (C t ) t∈T are reduced, irreducible and of dimension one. We say that the family dominates X if ∪ t∈T C t is dense in X. We say that the family avoids small sets if, given any small set S ⊂ X, there exists a dense open

T • ⊂ T such that C t ∩ S = ∅, for all t ∈ T • . Definition 2.3 (Pair). A pair (X, ∆) consists of a normal variety X and a Q-Weil divisor ∆ on X with coefficients in [0, 1] ∩ Q. A pair (X, ∆) is called snc if X is
smooth and if the support of ∆ has simple normal crossings only. We denote the maximal open subset of X where (X, ∆) is smooth by (X, ∆) snc . Note that this is a big subset of X. The fractional part of ∆ is written as {∆}.

Notation 2.4 (Reflexive hull). Given a normal, quasi-projective variety X and a coherent sheaf E on X, write

Ω [p] X := Ω p X * * , E [m] := E ⊗m * * and det E := ∧ rank E E * * .
Given any morphism f : Y → X, write f [ * ] E := ( f * E ) * * , etc.

Morphisms.

Let us now briefly review some basic notions and properties of morphisms between normal varieties.

Construction 2.5 (Push-forward of Weil divisor). Let f : X → Z be a morphism of normal varieties. Recall from "Zariski's Main Theorem in the form of Grothendieck"2 that there exists a unique, normal variety X and a unique factorisation

f = β • α as follows, X α open immersion / / X β proper morphism / / Z.
Taking Zariski-closures yields a push-forward morphism α * : WDiv(X) → WDiv(X). Composing with the standard push-forward morphism of the proper morphism β, cf. [Ful98, I Sect. 1.4], we obtain a map f * : WDiv(X) → WDiv(Z).

Remark 2.6 (Push-forward vs. linear equivalence). The map f * of Construction 2.5 will in general not respect linear equivalence, unless one of the following folds.

(2.6.1) The morphism f is proper.

(2.6.2) The variety X is a big open subset of Z and f is the inclusion. In this case, f * is an isomorphism. Warning 2.9 (Galois ⇒ étale). Definition 2.8 does not require γ to be étale. This will be of crucial importance for nearly everything that follows.

2.3. Equidimensional morphisms. In a normal variety, Weil divisors need not be Cartier. Still, it is possible to define a pull-back map, at least for equidimensional morphisms.

Definition 2.10 (Equidimensional morphism). Let f : X → Z be a dominant morphism of varieties. We say that f is equidimensional if there exists a number d such for any x ∈ X, the associated fibre f -1 f (x) is of pure dimension d. The number d is called relative dimension.

Remark 2.11 (Preimages of big and small sets). In the setting of Definition 2.10, if

Z ′ ⊆ Z is any algebraic set, then codim X f -1 (Z ′ ) ≥ codim Z Z ′ .
In particular, preimages of small sets are small, and preimages of big sets are big.

Lemma 2.12 (Equidimensional morphism and normalisation). Let f : X → Z be a dominant, equidimensional morphism of varieties. If X is normal and Z ′ the normalisation of Z, then the natural morphism f ′ : X → Z ′ is likewise equidimensional.

Proof. Recalling that the normalisation morphism η : Z ′ → Z is finite, it follows that for any x ∈ X, the fibre

F ′ := ( f ′ ) -1 f ′ (x) is a union of connected components of F = f -1 f (x). If F is of pure dimension d, then so is F ′ .
2.3.1. Pull-back. We now explain the construction of pull-back maps for Weil divisors in a normal variety.

Construction 2.13 (Pull-back of Weil divisor). Let f : X → Z be an equidimensional morphism between normal varieties. We define a pull-back morphism f * : WDiv(Z) → WDiv(X) as the composition of the following morphisms, WDiv(Z)

∼ = -------------→ by (2.6.2) since Z reg is big WDiv(Z reg ) ∼ = -→ CDiv(Z reg ) f | f -1 (Zreg) * --------→ CDiv f -1 (Z reg ) → WDiv f -1 (Z reg ) ∼ = -------------→

WDiv(X).

Since all morphisms respect linear equivalence, so does f * .

Multiplicities, ramification and branch divisors.

We briefly review various notions of multiplicities that appear in the following sections.

Definition 2.14 (Multiplicities). Let f : X → Z be an equidimensional morphism between normal varieties. If ∆ ∈ WDiv(X) is prime, define the multiplicity of f along D as mult ∆ f := mult ∆ f * D, where D := ( f * ∆) red .

Remark 2.15. In Definition 2.14, either supp ∆ dominates Z, or f * ∆ = 0 and mult ∆ f ≥ 1.

Definition 2.16 (Ramification and branch divisors). Let f : X → Z be an equidimensional morphism between normal varieties. The ramification and branch divisor of f are defined as follows,

Ramification f := ∑ ∆∈WDiv(X) prime max 0, mult ∆ f -1 • ∆ Branch f := ∑ D∈WDiv(Z) prime D≤ f * Ramification f lcm{mult ∆ f * D | ∆ ⊆ supp f * D a prime div.} • D OrbiBranch f := ∑ D∈WDiv(Z) prime D≤Branch f 1 -mult D Branch f mult D Branch f • D Remark 2.
X • \ (supp Ramification f ) → Z • X • ∩ (supp Ramification f ) → Z • ∩ (supp Branch f ).
Remark 2.17 ensures that the second map is defined. Observe that both Z • and X • are big open subsets of Z and X, respectively. Let x ∈ X • be any point and z := f ( x) ∈ Z • be its image. If x is not contained in the support of Ramification( f ), then f is smooth at x. If z 0 , . . . , z n ∈ O Z, z are local holomorphic coordinates on Z centred about z, then x i := z i • f ∈ O X, x can be completed to a system of holomorphic coordinates on X centred about x. In these coordinates, f takes the form (2.18.1) f : (x 0 , . . . , x n , x n+1 , . . . , x n+d ) → (x 0 , . . . , x n ).

If x is contained in the support of Ramification( f ), then there exists a holomorphic function z 0 ∈ O Z, z which locally generates the ideal of the smooth hypersurface supp Branch( f ) . Near x, there exists a holomorphic function

x 0 ∈ O X • , x such that z 0 • f = x m
0 , where m is the order of ramification of f along the unique component of Ramification( f ) that contains x. Completing, we obtain holomorphic coordinate functions of the following form, z 0 , . . . , z n ∈ O Z, z and x 0 , x 1

:=z 1 • f , . . . , x n :=z n • f , x n+1 , . . . , x n+d ∈ O X, x .
In these coordinates, f takes the form (2.18.2) f : (x 0 , x 1 , . . . , x n , x n+1 , . . . , x n+d ) → (x m 0 , x 1 , . . . , x n ). Notation 2.19 (Local normal form). In the setting of Construction 2.18, we refer to the explicit description of f in (2.18.1) and (2.18.2) as local normal forms. We call X • and Z • the maximal open sets where f can locally be written in normal form. If X • = X, we say that f can locally be written in normal form.

Rational maps.

A certain class of rational maps and divisors appear naturally in the proof of the semipositivity theorem, Theorem 5.3. The current section is devoted to introducing these maps and reviewing some of their basic properties.

Notation 2.20 (Domain of definition, preimages, connected fibres). Let f : X Z be a rational map between varieties. We denote the domain of definition of Defn( f ) ⊆ X and write f Defn for the morphism f | Defn( f ) . Given any subset

Z ′ ⊆ Z, write f -1 (Z ′ ) as a shorthand for f -1 Defn (Z ′ ). If z ∈ Z is any point, call X z := f -1 (z)
f := Ramification f Defn | U .
Ditto for Branch f and OrbiBranch f . 2.4.1. Pull-back and push-forward. In the rest of the current subsection we focus on the behaviour of cycles and sheaves under f * and f * , given that f is a rational map between normal varieties. Construction 2.25 (Push-forward for rational map). Let f : X Z be a rational map between normal varieties. Since Defn( f ) is a big subset of X, we obtain a canonical identification WDiv Defn( f ) ∼ = WDiv(X). Construction 2.5 therefore gives a push-forward map f * : WDiv(X) → WDiv(Z).

Construction 2.26 (Pull-back of sheaves). Let f : X Z be a rational map between normal varieties. If F is any coherent sheaf of O Z -modules, write

f [ * ] F := ι * f * Defn F * * ,
where ι : Defn( f ) → X is the inclusion. Since Defn( f ) is a big, open set, this is a coherent, reflexive sheaf on X.

Construction 2.27 (Pull-back of divisors for essentially equidimensional map). Let f : X Z be an essentially equidimensional rational map. Item (2.6.2) of Remark 2.6 gives a canonical identification WDiv(U) ∼ = WDiv(X) that respects linear equivalence. Construction 2.13 therefore gives a pull-back map f * : WDiv(Z) → WDiv(X), which does not depend on the choice of U, respects linear equivalence, and therefore induces a morphism between divisor class groups.

Remark 2.28 (Pull-back for Weil divisorial sheaves). The pull-back Constructions 2.26 and 2.27 are compatible for Weil divisorial sheaves. More precisely, if f : X Z is any essentially equidimensional rational map between normal varieties and if D ∈ WDiv(Z), then

f [ * ] O Z (D) ∼ = O X f * D .
2.4.2. Relative tangent sheaves. The aim of this section is to establish an explicit description for the relative canonical sheaf of an essentially equidimensional rational map.

Construction 2.29 (Relative tangent sheaf). Let f : X Z be an essentially equidimensional rational map between normal varieties. Recall from Remark 2.11 that there exists a big, open set U ⊆ f -1 (Z reg ) ∩ X reg such that f | U : U → Z reg is an equidimensional morphism. Denote the inclusion by ι : U → X, consider the kernel

T U/Z reg := ker T U → ( f | U ) * T Z reg ),
and set T X/Z := ι * T U/Z reg . By construction, T X/Z is a reflexive subsheaf of T X , and in fact a foliation (see Notation 2.32 below). The sheaf T X/Z is independent of the choice of U. Construction 2.30 (Relative canonical class). Let f : X Z be an essentially equidimensional rational map. Construction 2.27 allows to define the relative canonical class as

[K X/Z ] := [K X ] -f * [K Z ] ∈ Cl(X).
Lemma 2.31 (Determinant of relative tangent sheaf). In the setting of Construction 2.29,

det T X/Z ∼ = O X -K X/Z + Ramification f .
Proof. Since both sides of the equation are reflexive sheaves, it suffices to show equality on the big, open subset of U where the morphism f | U can locally be written in normal form. There, the claim follows from an elementary computation in coordinates.

2.5. Foliations. The notion of a foliation being transversal to a divisor is a recurrent theme in this chapter. Let us briefly spell out what is meant by this.

Notation 2.32 (Foliation). Let X be a normal variety. A foliation is a saturated subsheaf T X , whose restriction to X reg is closed under the Lie-bracket.

Remark 2.33. In the setting of Notation 2.32, recall that the tangent sheaf T X := Ω 1 X * is reflexive. As a saturated subsheaf, the foliation F is likewise reflexive. Notation 2.34. For any saturated subsheaf F of T X , the Lie-bracket, which is defined only on X reg , induces an O X -linear map

N : F [2] → (T X /F ) * *
known as the O'Neil tensor. Vanishing of N characterises then F as being a foliation.

Notation 2.35 (Divisors generically transveral to a foliation). In the setting of Notation 2.32, there exists a big open set U ⊆ X, contained in X reg where F is a subvectorbundle of T X . If D ⊂ X is any prime divisor, then F is as subvectorbundle of T X near general points x ∈ D. In particular, one can check whether F is transversal to D at x. This allows to decompose any Q-Weil divisor ∆ as ∆ = ∆ trans + ∆ ntrans , where ∆ trans consists of those components that are generically transversal to F . Remark 2.36. In the setting of Construction 2.29, the decompositions of Q-Weil divisors given in Notations 2.21 and 2.35 agree. More precisely, if ∆ is any Q-Weil divisor on X, then ∆ horiz = ∆ trans and ∆ vert = ∆ ntrans . 2.6. Adapted morphisms. In this section we introduce a class of morphisms, called "adapted", that is indispensable to even formulate our main result. Definition 2.37 (Adapted and strongly adapted cover cover). Let (X, ∆) be a pair and decompose ∆ into irreducible components, ∆ = ∑ δ j D j . Let γ : Y → X be an essentially equidimensional morphism to a normal variety Y. The morphism γ is called adapted to (X, ∆) if γ * ∆ is an integral divisor. The morphism γ is called strongly adapted to (X, ∆) if for any index j with δ j ∈ (0, 1), written as δ j = a j b j with a j , b j ∈ N coprime, and any Weil divisor E Y, we have

mult E (γ * D j ) ∈ {0, b j }.
In other words, if E appears in γ * D j at all, then its multiplicity must be b j precisely.

2.6.1. Existence. The existence of adapted morphisms, when (X, ∆) is an snc pair, was established by Kawamata, cf. [Laz04a, Prop. 4.1.12]. Here we briefly review the case where (X, ∆) is not snc. See [CP15b, Subsect. 1.2] for an alternative con- struction.

Proposition 2.38 (Existence of strongly adapted, Galois covers). Let (X, ∆) be a pair. Then, there exists a cyclic Galois cover γ : Y → X that is strongly adapted to (X, ∆).

Proof. Let π : X → X be a log resolution of singularities, and consider the Q-Weil divisor given as the strict transform, D := γ -1 * ∆. The existence of cyclic Galois cover γ : Y → X that is strongly adapted to ( X, D) has been recalled in [JK11a, Prop. 2.9]. We obtain the desired covering by Stein factorisation of the composed morphism Y → X,

Y α / / γ•π ) ) X β / / X.
The equivariant version of Zariski's Main Theorem, [GKP16, Theorem A.1], guarantees that β is Galois, and that its Galois group equals that of γ.

2.6.2. Relation to earlier definitions. Definition 2.37 is equivalent to various other definitions of adapted morphisms that appear in the literature -it goes without saying that all are various takes on the original definition of Campana. To see this, it is convenient to first introduce the following definition of the round-up of a Q-divisor. Definition 2.39 (C-round-up). Let (X, ∆) be a pair and decompose ∆ into irreducible components, ∆ = ∑ δ j D j . If j is any index with δ j ∈ (0, 1), write δ j = a j b j with a j , b j ∈ N coprime. Finally, set

⌈δ j ⌉ C =        0 if δ j = 0 1 if δ j = 1 b j -1 b j otherwise.
We call the divisor ⌈∆⌉

C := ∑ j ⌈δ j ⌉ C • D j the C-round-up of ∆. If ∆ = ⌈∆⌉ C , we call (X, ∆) a C-pair.
Remark 2.40 (Comparison with earlier definitions). Let (X, ∆) be a pair and γ : Y → X a covering map. Then the following are equivalent.

(2.40.1) The cover γ is adapted to (X, ∆) in the sense of Definition 2.37.

(2.40.

2) The cover γ is adapted to (X, ⌈∆⌉ C ) in the sense of Definition 2.37.

(2.40.

3) The morphism γ is adapted to (X, ⌈∆⌉ C ) in the sense of [JK11a, Defini- tion 2.7]. Ditto for strongly adapted covers. 2.7. Numerical classes, positivity. Over a Q-factorial projective variety the determinant of any coherent sheaf naturally defines an element of N 1 (X) Q . To avoid potentially cumbersome notations, let us fix a notation for such numerical classes. Notation 2.41 (Numerical classes). Let X be a Q-factorial, projective variety and F a coherent sheaf of O X -modules. Consider the Weil divisorial sheaf det F := (∧ rank F F ) * * -when F is torsion and its rank is zero, then det F is nothing but the zero sheaf. The numerical class of det F will be written as

[F ] ∈ N 1 (X) Q .
Warning 2.42 (Lack of additivity). Note that the numerical class operator [•] is not necessarily additive in exact sequences. In fact, since the reflexive hull of any torsion sheaf is zero, the ideal sheaf sequence of any Cartier divisor will give a counterexample.

Notation 2.43 (Harder-Narasimhan filtration, generic positivity). Let X be a normal, projective variety and H be an ample Cartier divisor on X. If F is any torsion free, coherent sheaf of O X -modules, consider the associated Harder-Narasimhan filtration 0

= F 0 F 1 • • • F r = F . With this notation in place, write µ max H (F ) := µ H (F 1 ) and µ min H (F ) := µ H (F r /F r-1
). We call F generically semipositive with respect to H if µ min H (F ) ≥ 0. We call F generically semipositive if F is generically semipositive with respect to any ample divisor.

Part I. Fractional semipositivity and application to hyperbolicity

LOGARITHMIC DIFFERENTIALS WITH FRACTIONAL POLE ORDER

In this section we define the sheaves of adapted differential forms. These sheaves are, in a sense, the natural generalisation of sheaves of log-differential forms for pairs (X, D) with reduced boundary divisor D, to the context of pairs (X, ∆) with

∆ = ∑ δ i • ∆ i , where δ i ∈ [0, 1] ∩ Q are fractional.
Their construction depends on the choice of the adapted morphism. Campana realised that, even in the purely logarithmic setting of Viehweg's hyperbolicity problem, they provide great flexibility in dealing with birational problems. We begin this section by explaining the local description of these sheaves when (X, ∆) is snc.

Informal explanation and local computation.

Throughout the present Section 3.1, we consider the following particularly simple setting.

Setting 3.1 (Setup and notation for Section 3.1). Let (X, ∆) be an an snc pair. Let γ : Y → X be a cover that is adapted to (X, ∆) and can locally be written in normal form. Assume that supp(∆ + Branch γ) and supp(γ * ∆ + Ramification γ) are both smooth. Choose a point y ∈ Y and set x := γ(y). Observe that if x ∈ supp ∆, then there exists exactly one component of D ⊆ ∆ that contains x. Let δ be the coefficient of this component. If x ∈ supp ∆, set δ := 0.

We choose coordinates x 0 , . . . , x n ∈ O X, x and y 0 , . . . , y n ∈ O Y, y , centred about x and y, respectively, that present γ in local normal form. In particular, there exists a number m such that x 0 • γ = y m 0 . If δ = 1 and if γ happens to be unramified at y, we may assume that locally near x, the divisor D is given as {x 0 = 0}. We are interested in writing formal fractional-exponent-differential forms σ := x -δ 0 • dx 0 . While this is not well-defined on X, one can write down the formal pullback of σ to Y; this will lead to the definition of adapted differentials. For the convenience of the reader, we discuss the cases where δ = 0, where 0 < δ < 1 and where δ = 1 separately.

3.1.1. The case where 0 < δ < 1. In this case, the divisor D is necessarily contained in the branch locus of γ, and locally given as {x 0 = 0}. One may formally write

(dγ)(σ) = y -m•δ 0 • d(y m 0 ) = m • y (m-1)-mδ 0 • dy 0 .
The assumption that γ is adapted ensures that m • δ is integral, and hence so is the exponent of y 0 . The fact that δ < 1 ensures that the exponent is not negative. We aim to define a sheaf of adapted differentials, in symbols Ω 1 (X,∆,γ) , as a subsheaf of Ω 1 Y , whose stalk at y is generated by the forms y (m-1)-mδ 0

• dy 0 and dy 1 , . . . ,

dy n ∈ Ω 1 Y, y .
Warning 3.2. It might seem tempting to take this as a definition for the sheaf of adapted differentials. However, the following example shows that this is quite delicate. Let Z be a smooth variety and H a smooth hypersurface on Z. Let z ∈ H be any and z 0 , . . . , z n ∈ O Z, z a regular system of parameters, where z 0 generates the ideal of H. Then note that the span

z 2 0 • dz 0 , dz 1 , . . . , dz n ⊂ Ω 1 Z, z
does depend in a non-trivial way on the choice of coordinates. To give a proper definition, it will always be necessary to take the morphism γ into account.

In order to define Ω 1 (X,∆,γ) properly, in a coordinate-free way, we compare its set of generators-to-be to the well-known set of generators for the image of the pull-back map dγ :

γ * Ω 1 X → Ω 1 Y , Ω 1 (X,∆,γ), y = y (m-1)-mδ 0 • dy 0 , dy 1 , . . . , dy n ⊆ Ω 1 Y, y Image(dγ) y = y m-1 0 • dy 0 , dy 1 , . . . , dy n ⊆ Ω 1 Y, y .
This suggests to define Ω 1 (X,∆,γ) near the point y in one of the two following, equivalent ways,

Ω 1 (X,∆,γ) := Ω 1 Y ⊗ y (m-1)-mδ 0 + Image(dγ) (3.2.1) Ω 1 (X,∆,γ) := Image(dγ) ⊗ y -mδ 0 ∩ Ω 1 Y . (3.2.2) Explanation 3.3 (Ideals in (3.2.1)). In (3.2.1), we view y (m-1)-mδ 0 as an ideal, view Ω 1 Y ⊗ y (m-1)-mδ 0
as a subsheaf of Ω 1 Y , and the sum is the sum of coherent subsheaves there.

Explanation 3.4 (Weil divisorial sheaves in (3.2.2)). In (3.2.2), we view y -mδ 0 as the Weil divisorial sheaf generated by the rational function y -mδ 0 , view Ω 1 Y as a subsheaf of Image(dγ) ⊗ y -mδ 0 , and the intersection is the intersection of coherent subsheaves there.

In order to avoid the awkward use of adapted coordinates, observe that the divisor given by y m-1 0 equals the ramification divisor of γ, while the divisor given by y mδ 0 is the pull-back divisor γ * ∆. Definitions (3.2.1)-(3.2.2) thus simplify as follows,

Ω 1 (X,∆,γ) := Ω 1 Y ⊗ O Y -γ * ∆ -Ramification(γ) + Image(dγ) (3.4.1) Ω 1 (X,∆,γ) := Image(dγ) ⊗ O Y (γ * ∆) ∩ Ω 1 Y . (3.4.2)
3.1.2. The case where δ = 0. In this case, γ may or may not be ramified at y. The form σ = x -δ 0 • dx 0 = dx 0 is an ordinary differential form, and so is its pull-back (dγ

)(σ) = m • y m-1 0 • dy 0 . We would then set Ω 1
(X,∆,γ) := Image(dγ) near the point y. Observe that this definition agrees with (3.4.1)-(3.4.2) above.

3.1.3. The case where δ = 1. In this case, γ may or may not be ramified at y. If γ is ramified at y, then the assumption that supp(∆ + Branch γ) is smooth implies that near x, the divisor D equals the branch locus, and is given as {x 0 = 0}. The form σ = x -1 0 • dx 0 = d log x 0 is a logarithmic differential form, and so is its pullback (dγ)(σ) = d log y 0 . In this case, we would like to define the sheaf of adapted differentials near y as

Ω 1 (X,∆,γ) := Ω 1 Y (log ∆ γ ) = γ * Ω 1 X (log⌊∆⌋)
, where ∆ γ := (γ * ⌊∆⌋) red . Formulas (3.4.1)-(3.4.2) include this case after the following minor adjustment. In fact, extending the pull-back morphism dγ to include logarithmic differentials, dγ :

γ * Ω 1 X (log⌊∆⌋) → Ω 1 Y (log ∆ γ ), we can write Ω 1 (X,∆,γ) := Ω 1 Y (log ∆ γ ) ⊗ O Y -γ * {∆} -Ramification(γ) + Image(dγ) Ω 1 (X,∆,γ) := Image(dγ) ⊗ O Y γ * {∆} ∩ Ω 1 Y (log ∆ γ )
. These formulas will re-appear in the succeeding Section 3.2, where adapted differentials are formally introduced.

Formal definition.

We now give a formal and coordinate-free definition of "adapted differentials", following the discussion of the previous subsection. A local description is also included.

3.2.1.

Adapted differentials for a good cover. We define adapted differentials first for covers that satisfy all the assumptions of Setting 3.1. We call such covers good. To be more precise, the following definition will be used. Definition 3.5. Let (X, ∆) be a pair, and γ : Y → X be a cover. The cover is called good if the following properties hold.

(3.5.1) The variety X and its subvariety supp(∆ + Branch γ) are smooth.

(3.5.

2) The variety Y and its subvariety supp(γ * ∆ + Ramification γ) are smooth.

(3.5.3) The cover γ is adapted.

(3.5.4) The cover γ can locally be written in normal form.

As indicated above, we take (3.4.2) as the definition of adapted differentials, at least for good covers. Definition 3.6 (Adapted differentials for good cover). Let (X, ∆) be a pair, and γ : Y → X a be good cover. Consider the pull-back map of logarithmic differentials,

dγ : γ * Ω 1 X log⌊∆⌋ → Ω 1 Y (log ∆ γ )
, where ∆ γ := (γ * ⌊∆⌋) red . The sheaf of adapted differentials on Y is then defined as

Ω 1 (X,∆,γ) := Image(dγ) ⊗ O Y γ * {∆} ∩ Ω 1 Y (log ∆ γ ),
where the intersection is the intersection of subsheaves in Ω 1 Y (log ∆ γ ) ⊗ O Y (γ * {∆}). Remark 3.7 (Inclusions between sheaves of adapted differentials). In the setting of Definition 3.6, it follows immediately from the definition that there exist inclusions, (3.7.1)

γ * Ω 1 X (log⌊∆⌋) ⊆ Ω 1 (X,∆,γ) ⊆ Ω 1 Y log ∆ γ ,
satisfying the following properties.

(3.7.

2) The first inclusion in (3.7.1) is an equality away from supp γ * {∆}.

(3.7.

3) The three terms of (3.

γ * Ω 1 X = y m-1 0 • dy 0 , dy 1 , . . . , dy n Ω 1 (X,∆,γ) = y (m-1)-mδ 0
• dy 0 , dy 1 , . . . , dy n Ω 1 Y = dy 0 , dy 1 , . . . , dy n .

The following is now an immediate consequence of the local description.

Corollary 3.9 (Determinants and Chern classes of adapted differentials). In the setting of Definition 3.6, we have equalities of sheaves,

det Ω 1 (X,∆,γ) = det γ * Ω 1 X (log⌊∆⌋) ⊗ det O Y γ * {∆} = O Y γ * (K X + ⌊∆⌋) + γ * {∆} = O Y γ * (K X + ∆) . In particular, c 1 Ω 1 (X,∆,γ) = [γ * (K X + ∆)].
3.2.2. Adapted reflexive differentials in the general setting. We now extend the definition of the adapted differentials from good covers to arbitrary ones.

Definition 3.10 (Adapted reflexive differentials). Let (X, ∆) be a pair, and let γ : Y → X be a cover that is adapted to (X, ∆). Let X

• := ∆| X • and γ • := γ| Y • .
We define the sheaf of adapted reflexive differentials on Y as

Ω [1] (X,∆,γ) := ι * Ω 1 (X • ,∆ • ,γ • ) .
where Ω 1

(X • ,∆ • ,γ • )
is the sheaf that has been introduced in Definition 3.6 on the facing page. Remark 3.11 (Inclusions between sheaves of adapted reflexive differentials). In the setting of Definition 3.10, observe that Ω 1 (X • ,∆ • ,γ • ) is locally free on Y • . It follows that the sheaf of adapted reflexive differentials is reflexive. Using (3.7.1), pushforward from Y • to Y induces inclusions of reflexive sheaves as follows, (3.11.1)

γ [ * ] Ω [1] X (log⌊∆⌋) ⊆ Ω [1] (X,∆,γ) ⊆ Ω [1]
Y log ∆ γ . where ∆ γ := (γ * ⌊∆⌋) red . Remark 3.7 and Corollary 3.9 also have direct analogues. (3.11.2) The first inclusion in (3.11.1) is an equality away from supp γ * {∆}.

[1] (X,∆,γ) = O Y γ * (K X + ∆) . If K X + ∆ is Q-factorial, then c 1 Ω 1 (X,∆,γ) = [γ * (K X + ∆)]. 3.2.3. Relation to earlier definitions. If (X, ∆) is a C-

FRACTIONAL TANGENTS AND FOLIATIONS

The aim of this section is to lay down the technical groundwork for Section 8. There, we construct a certain subsheaf of T X and study its integrability properties. The key technical result here is Proposition 4.5, whose proof requires some preliminary observations about the local description of vector fields that are transversal (resp. tangential) to the branch locus of an adapted cover.

4.1. Adapted tangents. We first define, in the obvious way, the notion of an adapted tangent sheaf, by dualizing the adapted sheaf of differentials.

Definition 4.1 (Adapted tangents). Given a pair (X, ∆) and an adapted cover

γ : Y → X, set T (X,∆,γ) := Ω [1] (X,∆,γ) * ,
where Ω

[1] (X,∆,γ) is the sheaf of adapted reflexive differentials that was introduced in Definition 3.10. We call T (X,∆,γ) the adapted tangent sheaf or sheaf of adapted tangents.

T Y -log ∆ γ ⊆ T (X,∆,γ) ⊆ γ [ * ] T X (-log⌊∆⌋).
where

∆ γ := (γ * ⌊∆⌋) red .
(X,∆,γ) = O Y -γ * (K X + ∆) . If K X + ∆ is Q-factorial, then c 1 T (X,∆,γ) = [-γ * (K X + ∆)].
4.1.1. Adapted tangents for good covers. For a good adapted cover, we were able to give a complete descriptions of adapted differentials in Section 3.2.1. The following is the direct analogue of this for adapted tangents.

Remark 4.3 (Local description of adapted tangents). Setup as in Definition 4.1. If the cover γ is good, then the inclusions (4.2.1) can be written down in local coordinates, near any given point y ∈ Y. If γ is étale at y, or if y ∈ supp ∆ γ then all three sheaves agree, and there is nothing much to do. Let us therefore assume that y ∈ Ramification( f ) \ supp ∆ γ . Choose local coordinates as in Setup 3.1 and follow the notation introduced there.

Near y, the sheaves γ * T X (-log⌊∆⌋) and γ * T X agree, and so do the sheaves T Ylog(γ * ⌊∆⌋) red and T Y . The sheaf γ * T X is freely generated as an O Ymodule by symbols γ * ∂ ∂x 0 , . . . , γ * ∂ ∂x n . The sheaves of (4.2.1) are then generated as follows, (4.3.1)

γ * T X = γ * ∂ ∂x 0 , γ * ∂ ∂x 1 , . . . , γ * ∂ ∂x n T (X,∆,γ) = y mδ 0 • γ * ∂ ∂x 0 , γ * ∂ ∂x 1 , . . . , γ * ∂ ∂x n T Y = y m-1 0 • γ * ∂ ∂x 0 , γ * ∂ ∂x 1 , . . . , γ * ∂ ∂x n .
The local description has the following consequence, which will be relevant in the study of foliations.

Lemma 4.4. In the setting of Definition 4.1, let V ∈ H 0 X, T X (-log⌊∆⌋) be any logarithmic vector field on X, with associated pull-back

γ * V ∈ H 0 Y, γ * T X (-log⌊∆⌋) .
Then, the following holds.

(4.4.1) If V is everywhere transversal to the smooth subvariety supp({∆} + Branch γ), then γ * V generates the quotient γ * T X (-log⌊∆⌋)/T (X,∆,γ) . (4.4.2) If V is everywhere tangential to the smooth subvariety supp({∆} + Branch γ), then

γ * V is contained in H 0 Y, T Y (-log ∆ γ ) .
Proof. Both items can be shown locally, near given points y ∈ Y. Again, we choose local coordinates as in Setup 3.1 and follow the notation introduced there. Write the vector field V and its pull-back locally as

V = n ∑ i=0 f i • ∂ ∂x i and γ * V = n ∑ i=0 ( f i • γ) • γ * ∂ ∂x i where f i ∈ O X, x .
Proof of (4.4.1). If y is not contained in supp γ * {∆}, then we have seen in Item (4.2.2) of Remark 4.2 that T (X,∆,γ) = γ * T X (-log⌊∆⌋), so there is nothing to show. Let us therefore assume that y ∈ supp γ * {∆}. This allows to use the local description of adapted tangent from Remark 4.3. The assumption that V is transversal implies that f 0 does not vanish at x. But then f 0 • γ will not vanish at y, and the explicit description in (4.3.1) yields the claim.

Proof of (4.4.2). If y is contained in supp ∆ γ or in the complement of supp Ramification( f ), then we have seen in Item (4.2.3) of Remark 4.2 that the three sheaves of (4.2.1) agree, and there is nothing to show. We will therefore assume without loss of generality that ⌊∆⌋ = 0, and that y ∈ Ramification( f ).

The assumption that V is tangential implies that f 0 vanishes along {x 0 = 0}, so that f 0 = x 0 • g 0 and

γ * V = y m 0 • (g 0 • γ) • γ * ∂ ∂x 0 + n ∑ i=1 ( f i • γ) • γ * ∂ ∂x i .
Again, a comparison with the explicit description in (4.3.1) yields the claim.

4.2. Foliations. In this subsection we use the local machinery developed in Subsection 4.1 to establish a technical tool that will play a significant role in the proof of Theorem 5.3.

Lifting the O'Neil tensor.

A key problem in the proof of the main semipositivity result for a given pair (X, ∆) is to relate the integrability of a certain subsheaf F of T X to the behaviour of the pull-back of the O'Neil tensor3 on γ [ * ] F and G γ := (γ [ * ] F ∩ T (X,∆,γ) ). In the next proposition we show that the restriction of the lift of the O'Neil tensor maps the smaller sheaf G γ to T (X,∆,γ) /G γ , after taking reflexive hulls.

Proposition 4.5 (Lifting the O'Neil tensor). Let (X, ∆) be a pair and let γ : Y → X be an adapted cover. Let F ⊆ T X (-log⌊∆⌋) be a saturated subsheaf. Consider reflexive sheaves on G Y , G γ and G on Y as follows, (4.5.1)

T Y -log ∆ γ contains G Y := G ∩T Y ⊆ T (X,∆,γ) contains G γ := G ∩T Y ⊆ γ [ * ] T X (-log⌊∆⌋) contains G := γ [ * ] F . Next, consider the O'Neil tensor N : F [2] → T X (-log⌊∆⌋) F * *
, its reflexive pull-back,

γ [ * ] N : G [2] → γ [ * ] T X (-log⌊∆⌋) F ,
and write N γ for the restriction of γ

[ * ] N to the subsheaf G [2]
γ ⊆ G [2] . Then N γ factorises as follows,

G [2] γ α / / N γ + + T (X,∆,γ) G γ * * β / / γ [ * ] T X (-log⌊∆⌋) F .
Proof. This is easier than the involved notation suggests. An elementary diagram chase shows that the natural morphism β is injective.

Step 1: Simplification. To prove that a morphism of reflexive sheaves factorises via a third, it suffices to prove the existence of a factorisation on a big open set. We may therefore assume that the following additional properties hold. This simplifies notation greatly, as all sheaves in question will be locally free, so there is no need to take reflexive hulls in each step. There is more we can do. Recalling from Item (4.2.3) of Remark 4.2 that the three terms of (4.5.1) are equal near general points of supp ∆ γ , so that the claim of the proposition is certainly true there, we can also assume that the following holds.

(4.5.6) The integral part of ∆ is empty, so ⌊∆⌋ = 0 and ∆ γ = 0. Again, this simplifies notation substantially, allowing us to drop all "log⌊∆⌋" and "log ∆ γ " from sheaves of tangents and differentials.

Step 2: reduction to the local The statement of the proposition is clearly local; it suffices to prove the factorisation in an analytic neighbourhood of any given point y ∈ Y. Again, if γ is étale at y, then Item (4.2.3) of Remark 4.2 that the three terms of (4.5.1) are equal, and there is nothing to show. We will therefore assume that γ is ramified at y. Set x = γ( y) and let D ⊆ supp(∆ + Branch γ) be the unique component that contains x -the component is unique because supp(∆ + Branch γ) is smooth by the assumption that γ is a good cover. Replacing X with a suitably small neighbourhood of x, if need be, we can assume that the following holds in addition.

(4.5.7) The sheaf F is free, say generated by global sections σ 1 , . . . , σ r . (4.5.8) The support of ∆ + Branch γ is irreducible, D = supp(∆ + Branch γ). (4.5.9) If F is everywhere transversal to D, then σ 1 is everywhere transversal to D.

Step 3: Proof in case that F is everywhere transversal to D. The following diagram summarises the sheaves in question.

0 / / G γ / / _ G / / _ G G γ / / η 0 0 / / T (X,∆,γ) / / γ * T X / / γ * T X T (X,∆,γ) / / 0
The morphism η is injective by construction. Lemma 4.4 asserts that it is also surjective. More is true: Item (4.4.1) even asserts that the image is generated by the class of the section γ * σ 1 . The snake lemma thus implies that the natural map

T (X,∆,γ) G γ → γ * T X G is isomorphic.
The question of factorisation is therefore void and Proposition 4.5 is shown in case that F is everywhere transversal.

Step 4: Proof in case that F is everywhere tangential to D. If F is everywhere tangential to D, then Lemma 4.4 asserts that G is already contained in T Y , so that the sheaves G , G γ and G Y are actually equal. The composed morphism µ,

G [2] Y O'Neil tensor on Y / / µ , , T Y G Y / / γ * T X G
clearly equals N γ over the open set where γ is étale. Since all sheaves in question are locally free, hence torsion free, this means that N γ equals the map µ everywhere. But µ factors as desired. This shows Proposition 4.5 in the last remaining case.

Chern classes.

When proving the main result of this chapter, we will need to pull-back foliations from the variety X to an adapted cover, saturate, intersect and keep track of how Chern classes change in the process. The following somewhat technical lemma summarises everything that we will need.

Proposition 4.6 (Chern classes of foliations). Let (X, ∆) be a pair and let γ : Y → X be an adapted cover. Further, let F ⊆ T X be a foliation. Write ∆ = ∆ trans + ∆ ntrans , as in Notation 2.35 and consider the following sheaves.

G := F ∩ T X (-log⌊∆⌋) . . . saturated subsheaf of T X (-log⌊∆⌋) H := γ [ * ] G . . . saturated subsheaf of γ [ * ] T X (-log⌊∆⌋) H (X,∆,γ) := H ∩ T (X,∆,γ) . . . saturated subsheaf of T (X,∆,γ)
The determinant sheaves are then related as follows.

det G = det(F ) ⊗ O X (-⌊∆ trans ⌋) * * (4.6.1) det H = det H (X,∆,γ) ⊗ O Y γ * {∆ trans } + effective * * (4.6.2)
In summary,

γ [ * ] det F = det H (X,∆,γ) ⊗ O Y γ * ∆ trans + effective * * . (4.6.3)
Proof. Equation (4.6.1) is elementary. Equation (4.6.3) is a combination of (4.6.1) and (4.6.2). It remains to prove (4.6.2). As an equation between reflexive sheaves, (4.6.2) can therefore be checked on a big open set. We may therefore assume that γ is a good cover. Recalling from Item (4.2.2) of Remark 4.2 that the sheaves T (X,∆,γ)

and γ [ * ] T X agree away from the support of γ * {∆}, it will suffice to understand the difference between H and its subsheaf H (X,∆,γ) near a given point y in the support of γ * {∆ trans }. There, the statement follows from the local description (4.3.1) of Remark 4.3, using Item (4.4.1) of Lemma 4.4 as we have done in the proof of Proposition 4.5.

FRACTIONAL SEMIPOSITIVITY

A celebrated result of Miyaoka, [Miy87, Cor. 8.6], shows that for a smooth projective variety X (or more generally a normal projective variety with only canonical singularities) positivity properties of the canonical sheaf ω X are deeply related to those of the sheaf of (pluri-)differential forms. More precisely, if K X is pseudoeffective, then, for every positive integer m and ample divisor H ⊂ X, the sheaf (Ω 1 X ) ⊗m is semipositive with respect to H. In other words,

c 1 (Q) • [H] n-1 ≥ 0, for all coherent quotients Q of (Ω 1 X ) ⊗m .
When c 1 (X) = 0 or < 0, these results can be traced back to Yau's theorem on the existence of Kähler-Einstein metrics, [Yau77]. Miyaoka's approach, on the other hand, is purely algebraic and involves deep and delicate characteristic p methods. Campana and Pȃun extend Miyaoka's results to the context of pairs. Before we state their result in Theorem 5.3 below, we recall the definition of generic semipositivity with respect to an adapted cover. Definition 5.1 (Generic semipositivity w.r.t. an adapted cover, cf. [CP15b, Def. 1.7]). Let (X, ∆) be a projective pair where X, and let γ : Y → X be an adapted morphism that is Galois with group G. We say that Ω

[1] (X,∆,γ) is γ-generically semipositive if it is generically semipositive with respect to any ample divisor of the form γ * (ample).

Remark 5.2 (Intersection with nef classes). Nef divisors are limits of ample divisors.

If Ω

[1] (X,∆,γ) is γ-generically semipositive, continuity of intersection products therefore implies that c

1 (Q) • [γ * N] n-1 ≥ 0, for all coherent quotients Q of (Ω [1]
(X,∆,γ) ) ⊗m and all nef divisors N ∈ Q Div(X).

Theorem 5.3 (Generic semipositivity of Ω

[1] (X,∆,γ) , cf. [CP15b, Thm. 2.1]). Let (X, ∆) be a log canonical, projective pair, and let γ : Y → X be an adapted cover that is Galois with group

G. If K X + ∆ is pseudo-effective, then Ω [1] (X,∆,γ) is γ-generically semipositive.
Part II of this chapter is devoted to a proof of Theorem 5.3. After some preparatory sections, the proof is given in Section 9 on page 28.

APPLICATION TO HYPERBOLICITY

As recalled in the introduction, the first step in proving Viehweg's hyperbolicity Conjecture 1.1 was carried out by Viehweg and Zuo in Theorem 1.2, where it was shown that the variation in a smooth family of canonically-polarised manifolds f • : X • → Y • manifests itself as a birationally-positive subsheaf L of pluri-log forms. The aim of this section is to use Theorem 5.3 to extract bigness of the log canonical divisor K Y + D from the positivity of L . This is the content of the following theorem of Campana and Pȃun. Theorem 6.1 (Existence of pluri-log-canonical forms, cf. [CP15b, Thm. 4.1]). Let (X, D) be a projective snc pair, where D is reduced. Let N ∈ N + be a positive integer and let

L ⊆ N Ω 1 X log(D) be an invertible subsheaf. If κ(X, L ) = dim X, then κ(K X + D) = dim X.
6.1. Preparation for the proof of Theorem 6.1. Before we sketch the proof of Theorem 6.1 we gather some technical details in the following lemma, whose proof is contained in the arguments of [CP15b, Sect. 4] or those of [Taj16, Thm 5.2] and [Taj16, Claim 5.2.1].

Lemma 6.2. Setting as in Theorem 6.1. If κ(X, L ) = dim X, then there exists a Qdivisor B D ∈ Q Div(X), a positive integer M, and for every m

≥ M a Q-divisor G m ∈ Q Div(X) such that the following holds. (6.2.1) The divisor G m is big. Its round-down ⌊G m ⌋ is zero. (6.2.2) There exists a Q-linear equivalence D + 1 m B D ∼ Q G m . (6.2.3) The divisors D + 1 m B D and G m have snc support. (6.2.4) The pair (X, G m ) is klt. The pair X, D + 1 m B D is dlt. (6.2.5) The divisor K X + D + 1 m B D is pseudo-effective.
6.2. Proof of Theorem 6.1. Let M be the number and let B D and (G m ) m∈N + be the divisors of Lemma 6.2. Choose an ample divisor A ∈ Div(X), and choose a divisor L ∈ Div(X) that represents L . Write n := dim X.

Remark 6.4. As L is big, Kodaira's lemma [Laz04a, Prop. 2.2.6] implies that for any sufficiently positive integer a, we have a • L ≥ A.

The next step sets the stage for the proof.

Step 1: Minimal models for (X, G m ). The properties listed in Lemma 6.2 allow us to use [BCHM10, Thm. 1.1] and to conclude that the log-minimal models program for the pairs (X, G m ) terminate. In other words, for every number m, there exists a birational map π m : X X ′ m consisting of a finite number of flips and divisorial contractions, such that

(X ′ m , G ′ m ) is klt and K X ′ m + G ′ m is nef, where G ′ m denotes the strict transform of G m .
We resolve the indeterminacies of π m by blowing up. More precisely, choose log resolutions µ m of the pair (X, D + B D ) that fit into a commutative diagram of birational maps and morphisms as follows,

X m µ m } } π m " " X π m , MMP for (X,D m ) / / _ _ _ _ _ _ _ _ _ _ _ _ _ X ′ m .
The following will then hold. Write

A m := µ * m A, L m = µ * m (L) and L m := µ * m (L ).
With this notation, the standard pull-back of logarithmic forms then gives an inclusion

L m ⊆ N Ω 1 X m log( D m + E m ).
Step 2: Volume estimates. As a second step in the proof of Theorem 6.1, we aim to bound the volume of (X ′ m , G ′ m ) from below. More precisely, the following two claims will be shown.

Claim 6.5. Set ∆ m := D m + 1 m B Dm + E m ∈ Q Div X m
and consider the number c := aN • n aN-1 ∈ N + . Then, the following inequalities hold for all m ≥ M, for every ample divisor H m ∈ Div(X ′ m ) and every r ∈ N + , (6.5.1)

c • K X m + ∆ m -a • L m • π * m (K X ′ m + G ′ m + 1 r H m ) n-1 ≥ 0.
Proof of Claim 6.5. Pseudo-effectivity of

K X + D + 1 m B D implies that the pull-back µ * m (K X + D + 1 m B D
) is likewise pseudo-effective, and Item (6.2.4) implies that so is K X m + ∆ m . In particular, Theorem 5.3 ("Generic semipositivity") applies to the pair ( X m , ∆ m ). To this end, consider a morphism γ m : Y → X m that is adapted to the divisor ∆ m . Recall from [Laz04a, Prop. 4.1.12] that we may assume that the the cover Y is good, in the sense of Definition 3.5. In particular, we may assume that the sheaf Ω 1

( X m ,∆ m , γ m )
of adapted differentials is locally free. Next, consider the exact sequence of sheaves

0 → F m -→ aN Ω 1 ( X m ,∆ m , γ m ) → Q m → 0,
where F m is the saturation of γ * m L ⊗a m inside the middle term. Thanks to Theorem 5.3 and Remark 5.2, the torsion free sheaf Q m verifies the inequality:

[Q m ] • γ * m π * m K X ′ m + G ′ m + 1 r H m big and nef n-1 ≥ 0.
Inequality (6.5.1) now follows from an elementary computation of the first Chern class of Q m , using the equation c 1 Ω 1 (X,∆,γ) = [γ * (K X + ∆)] found in Corollary 3.9. Claim 6.5 is thus shown. Claim 6.6. Setting and notation as in Claim 6.5. Then, the following inequalities hold for all m ≥ M, for every ample divisor H m ∈ Div(X ′ m ) and every r ∈ N + , (6.6.1)

c • vol K X ′ m + G ′ m + 1 r H m ≥ A m • π * m K X ′ m + G ′ m + 1 r H m n-1
.

Proof of Claim 6.6. The birational map π m is a contraction. In other words, its inverse π -1 m does not contract any divisors. As a consequence, we see that any divisor in X m which get contracted by µ m must also be contracted by π m . In other words, the µ m -exceptional set E m ⊆ X m is also π m -exceptional. Together with Item (6.2.2), this observation yields the following Q-linear equivalence, (6.6.2)

K X m + ∆ m ∼ Q π * m (K X ′ m + G ′ m ) ± ( π m -exceptional
). Claim 6.6 now follows by putting things together.

A m • π * m K X ′ m + G ′ m + 1 r H m n-1 ≤ a • L m • π * m K X ′ m + G ′ m + 1 r H m n-1 Remark 6.4 ≤ c • K X m + ∆ m • π * m K X ′ m + G ′ m + 1 r H m n-1
Inequality (6.5.1)

≤ c • π * m K X ′ m + G ′ m • π * m K X ′ m + G ′ m + 1 r H m n-1
Equation (6.6.2)

≤ c • vol K X ′ m + G ′ m + 1 r H m
Claim 6.6 is thus established.

Step 3: Application of Claim 6.6, end of proof. According to Teissier inequality for nef divisors, [Laz04a, Thm. 1.6.1], Inequality (6.6.1) implies that

c • vol K X ′ m + G ′ m + 1 r H m ≥ vol A m 1/n • vol K X ′ m + G ′ m + 1 r H m (n-1)/n . In other words, vol K X ′ m + G ′ m + 1 r H m ≥ c -n • vol A m .
By taking r → ∞, we find that

vol K X ′ m + G ′ m ≥ c -n • vol( A m
). On the other hand, we know thanks to the negativity lemma in the minimal models program, that vol(K

X ′ m + G ′ m ) = vol(K X + G m

POSITIVITY OF RELATIVE DUALISING SHEAVES

As we shall see in Section 8, the orbifold generic semipositivity result, Theorem 5.3, is proved by contradiction. More precisely, given a pair (X, ∆) with pseudo-effective K X + ∆, and after integrability considerations (Subsection 8.1), the existence of a subsheaf of T (X,γ,∆) with positive slope leads to an algebraic foliation on X. The negativity properties of the relative canonical sheaf of the rational map associated to this foliation brings about the required contradiction to the pseudo-effectivity assumption of K X + ∆.

Theorem 7.1 ([CP15b, Thm. 2.11]). Let f : X Z be a rational map with connected fibres between normal, projective varieties. Assume that f is essentially equidimensional4 , that X is Q-factorial, that there exists a Q-Weil divisor ∆ on X such that (X, ∆) is log canonical, and that K X + ∆ is pseudo-effective. If (C t ) t∈T is a family of curves that dominates X and avoids small sets5 , then

K X/Z + ∆ horiz -Ramification f • [C t ] ≥ 0, for all t ∈ T.
Remark 7.2 (Q-factoriality in Theorem 7.1). The assumption that X is Q-factorial is posed for notational convenience. It implies that the intersection numbers in the displayed formula are well-defined. Note, however, that almost all curves in the family (C t ) t∈T stay away from the singularities of X. Restricting to these curves, and replacing X with a suitable log-resolution, it is possible to obtain a more general result at the cost of additional and more complicated notation.

Theorem 7.1 is a consequence of positivity results for direct images of relative dualising sheaves, which we present in Theorem 7.3 in the form of a pseudoeffectivity result for the relative dualising sheaf. These results have a long history, cf. [H ö10] for a survey in the setting where ∆ = 0. In case where the coefficients of ∆ are of the form m-1 m , the relevant positivity results for the push-forward of O X m • (K X/Z + ∆) appear in [Lu02, Sect. 9] and [Cam04, Thm. 4.11]; the paper [BP08] approaches the case where ∆ is integral by analytic methods. The general case, where the coefficients of ∆ are arbitrary, is treated in [Fuj14, Thm. 1.1] and (again in the analytic setting) in [PT14, Cor. 5.2.1]. For notational convenience, we choose [START_REF] Fujino | Notes on the weak positivity theorems[END_REF] as our main reference.

Theorem 7.3 (Pseudo-effectivity of relative dualising sheaves). Let f : X → Z be a surjective morphism with connected fibres between smooth, projective varieties. Let ∆ be a Q-divisor with snc support on X such that (X, ∆) is log canonical. Assume further that the restriction of K X/Z + ∆ to the general f -fibre is is pseudo-effective. Then K X/Z + ∆ is pseudo-effective.

Proof. Choose a very ample prime divisor H on X that is general in its linear systems. For every sufficiently small rational number 0 < ε ≪ 1, the divisor ∆ + ε • H will then have snc support, the pair (X, ∆ + ε • H) will be log canonical and if [START_REF] Viehweg | Quasi-projective moduli for polarized manifolds[END_REF]Lem. 2.16]. Since the target is invertible, it follows immediately from the definition of "weakly positive" that the Q-divisor Rem. 7.6]. Conclude by taking the limit ε → 0. 7.1. Proof of Theorem 7.1. The proof of Theorem 7.1 essentially consists of two parts. Part one is focused on modifying the rational map f : X Z and its subsequent replacement by a morphism that fits the premise of Theorem 7.3. This is roughly the content of Step. 1-3 and finally Step. 4 (see Consequences. 7.8 and 7.9). In Step. 5 it then becomes evident that Theorem 7.1 is an immediate consequence of Theorem

F ⊆ X is any general f -fibre, then (K X/Z + ∆ + ε • H)| F will be big. Choose a number m ≫ 0 such that the divisor m • (K X/Z + ∆ + ε • H) is integral and such that E := f * O X m • (K X/Z + ∆ + ε • H) is of
f * E = f * f * O X m • (K X/Z + ∆ + ε • H) → O X m • (K X/Z + ∆ + ε • H) , cf. [
K X/Z + ∆ + ε • H is pseudo- effective, [Fuj14,
Step 1: Resolution and base change. Following the construction steps outlined below, we construct a commutative diagram of morphisms and maps as follows, / / Z Z.

(7.3.1) Choose a strong log resolution of the morphism f and the pair (X, ∆). We obtain a smooth variety X and birational morphism b : X → X from a normal variety that is isomorphic over Defn( f ) ∩ (X, ∆) snc , such that f := f • b is a morphism and such that the b-exceptional locus E b as well as E b + b -1 * ∆ are divisors with simple normal crossing support. (7.3.2) Consider the divisor OrbiBranch( f ) that was introduced in Definition 2.24 on page 7. Proposition 2.38 allows to choose a strongly adapted cover β : Z → Z that is associated with the pair Z, OrbiBranch( f ) . Since β is finite, Construction 2.13 allows to consider the pull-back divisor β * OrbiBranch( f ). (7.3.3) Choose a strong log resolution of the pair Z, β * OrbiBranch( f ) . We obtain a smooth variety Z and a birational morphism α : Z → Z that is isomorphic wherever Z, β * OrbiBranch( f ) is snc. Both the α-exceptional locus E α as well as E α + α -1 * β * OrbiBranch( f ) are divisors with simple normal crossing support. (7.3.4) Choose a strong log resolution of the fibre product Z × Z X. We obtain a smooth variety X. Composed with the projection to the second factor, the resolution yields a generically finite morphism a : X → X. Let E b•a ⊆ X be the union of those divisors that are contracted by b • a. Then, both E b•a as well as E b•a + a * (b -1 * ∆) are divisors with simple normal crossing support.

Step 2: Open sets and local normal forms. Let Z • ⊆ Z and X • ⊆ f -1 (Z • ) be the maximal open sets such that the following holds.

(7.4.1) The pairs (X

• := (β • α) -1 (Z • ), the morphism (β • α)| Z • : Z • → Z • is finite
and can locally be written in normal form.

(7.4.4) Setting X

• := b -1 (X • ), the morphism b • := b| X• : X• → X • is iso- morphic. In particular, f | X• : X• → Z • can
locally be written in normal form. (7.4.5) Setting X • := a -1 ( X• ), the morphism a • := a| X • : X • → X• is finite and can locally be written in normal form.

Observation 7.5. Recall from Construction 2.18 and from Zariski's main theorem that Z • and X • are big open sets of Z and X, respectively.

Step 3: Adjunction for the morphism b. Decompose the b-exceptional divisor E b into irreducible components, (E b i ) i∈I b . Since (X, ∆) is lc, the standard adjunction for the morphism b reads

K X + b -1 * ∆ = b * K X + ∆ + ∑ a i E b i , with all a i ≥ -1. The Q-divisor ∆ := b -1 * ∆ -∑ a i <0 a i E b i
is effective with coefficients from the interval [0, 1] ∩ Q, and has simple normal crossings support. The pair ( X, ∆) is thus log canonical, and

K X + ∆ = b * K X + ∆ psef by assumpt. + ∑ a i >0 a i E b i is again pseudo-effective. To end with Step 3, set ∆h := ∆ -b -1 * ∆ vert and observe that if F ⊆ X is a general f -fibre, then F is disjoint from the support of b -1 * ∆ vert .
The following is thus an immediate consequence.

Observation 7.6. We have ∆h | X• = (b • ) * ∆ horiz . The pair ( X, ∆h ) is log-canonical and the restricted divisor K X + ∆h | F is pseudo-effective.

Step 4: Adjunction for the morphism a. Using Items (7.4.3) and (7.4.4), the local normal form of (β • α)| Z • and f | X• , as well as the construction of β as a strongly ad- apted cover, an elementary computation in local coordinates gives the following Q-linear equivalence,

K X • / Z • ∼ Q (b • • a • ) * K X • /Z • -Ramification( f ) .
A similar equation holds for pairs. For a precise formulation, let ι : X • → X be the obvious inclusion map and set

∆ h := ι * (a • ) * ( ∆h ) ,
where the push-forward ι * is taken in the sense of Construction 2.5 and Remark 2.6. It follows from the construction of the morphism a in (7.3.4) that every component of ∆ h dominates Z, and that no component of ∆ h | X • is contained in the ramification locus of the finite morphism a • . Likewise, if F ⊆ X is any general fibre of f , it follows from construction that a is étale near F. The following are thus immediate consequences of Observation 7.6.

Consequence 7.8. The pair ( X, ∆ h ) is log-canonical and satisfies

K X/ Z + ∆ h | X • ∼ Q (b • • a • ) * K X • /Z • -Ramification( f ) . Consequence 7.9. The restricted divisor K X + ∆ h | F is pseudo-effective.
Step 5: End of proof. The family (C t ) t∈T avoids small sets by assumption. Together with Observation 7.5, this means that if t ∈ T is general, then the curve C t is contained in X • . By (7.4.5), its preimage

C t := (b • a) -1 (C t ) is a curve in X • whose components are movable curves in X. Since the cycle-theoretic push-forward (b • a) * ( C t ) is a positive multiple of C t , we obtain 0 ≤ K X/ Z + ∆ h • C t
Thm. 7.3 and Cons. 7.9

= (b • a) * K X/Z + ∆ hor -Ramification( f ) • C t Cons. 7.8 = const + • K X/Z + ∆ hor -Ramification( f ) • C t Proj. formula.
This finishes the proof of Theorem 7.1.

FAILURE OF SEMIPOSITIVITY, CONSTRUCTION OF MORPHISMS

As was mentioned earlier, a key component of Campana-Pȃun's proof of the generic semipositivity result is the observation that given a lc pair (X, D) and an adapted cover γ : Y → X, any subsheaf F (X,∆,γ) ⊆ T (X,∆,γ) that is maximally destabilising respect to γ * (ample) induces an algebraic foliation on X, whose leaves are often algebraic. This is the content of the next theorem.

Theorem 8.1 (From positive subsheaves to foliations, cf. [CP15b, Sect. 2]). Let (X, ∆) be a projective pair and let γ : Y → X be an adapted morphism that is Galois with group G. Assume that Ω

[1] (X,∆,γ) is not γ-generically semipositive. Then, there exists a normal variety Z, a dominant, essentially equidimensional rational map ψ : X Z, and a family (C t ) t∈T of curves in X such that the following holds.

(8.1.1) The family (C t ) t∈T dominates X and avoids small sets. (8.1.2) Given any t ∈ T, we have [T

X/Z ] • [C t ] > [∆ horiz ] • [C t ].
Remark 8.2. The family (C t ) t∈T avoids small sets, and its general members are thus contained in X reg . The intersection numbers of Item (8.1.1) are therefore welldefined, even if X is not necessarily Q-factorial.

8.1. Proof of Theorem 8.1. As before, the proof is subdivided into a number of relatively independent steps.

Step 1: Setup. By assumption, there exists a very ample divisor A on X such that Ω

[1] (X,∆,γ) is not generically semipositive with respect to the ample divisor A γ := γ * A. Let and C γ ⊂ Y be a a general complete intersection curve for the ample divisor A Y on Y, in the sense of Mehta-Ramanathan. Consider the Harder-Narasimhan filtration of the sheaf T (X,∆,γ) of adapted tangents with respect to A γ and let F (X,∆,γ) ⊆ T (X,∆,γ) denote the maximally destabilising subsheaf, which is saturated in T (X,∆,γ) and hence reflexive. By assumption, its slope is positive, µ A γ F (X,∆,γ) > 0. Remark 4.2 yields an inclusion

F (X,∆,γ) ⊆ γ [ * ] T X (-log⌊∆⌋).
We denote the associated saturation by F sat (X,∆,γ) ⊆ γ [ * ] T X (-log⌊∆⌋). Since F (X,∆,γ) ⊆ T (X,∆,γ) is itself saturated, we have an equality F (X,∆,γ) = F sat (X,∆,γ) ∩ T (X,∆,γ) . Observation 8.3 (Regularity along C γ ). The curve C γ is a general member in a dominating family of curves that avoids small sets. In particular, the following holds. Observation 8.5 (G-invariance). The divisor A γ is invariant under the action of the Galois group. As a consequence, it follows from the uniqueness of the Harder-Narasimhan filtration that the maximally destabilising subsheaf F (X,∆,γ) is a Gsubsheaf of T (X,∆,γ) , and also of γ [ * ] T X (-log⌊∆⌋).

In a similar vein, it follows from uniqueness of saturation that F sat (X,∆,γ) is a G-subsheaf of γ [ * ] T X (-log⌊∆⌋). Thus, by [GKPT15, Prop. 2.16], there exists a reflexive, saturated subsheaf F ⊆ T X (-log⌊∆⌋) such that F sat (X,∆,γ) = γ [ * ] F .

Notation 8.6. Let F sat ⊆ T X denote the saturation of F inside T X . The following diagrams summarise the situation, Step 2: Construction of a foliation. Next, we will show that the sheaf F sat is in fact a foliation. For this, Proposition 4.5, which describes the lifting the O'Neil tensor to an adapted cover, will be the key ingredient.

T Y -log ∆ γ ⊆ T (X,∆,γ) contains F (X,∆,γ) ⊆ γ [ * ] T X (-log⌊∆⌋) contains F sat (X,∆,γ) =γ [ * ]
Claim 8.8. The sheaf F sat ⊆ T X is closed under the Lie-bracket.

Proof of Claim 8.8. Closedness under Lie bracket can be checked on an open subset. It will therefore suffice to show that the subsheaf F ⊆ T X (-log⌊∆⌋), which agrees with F sat generically, is closed under the Lie-bracket. Equivalently, we need to prove that the O'Neil tensor

N : F [2] → T X (-log⌊∆⌋) F * *
vanishes. For this, recall from Proposition 4.5 that the restriction of its reflexive pull-back to F (X,∆,γ) gives a morphism N (X,∆,γ) : F

[2] (X,∆,γ) → T (X,∆,γ) F (X,∆,γ) * * .

Since γ [ * ] F and F (X,∆,γ) agree on a dense open set, it will be enough to show that N (X,∆,γ) vanishes. This is actually the case for slope reasons. We have the following inequalities:

µ min A γ F [2]
(X,∆,γ) = 2 • µ min A γ F (X,∆,γ) > µ min A γ (F (X,∆,γ) ) since µ min > 0 by assumption

≥ µ max A γ
T (X,∆,γ) F (X,∆,γ) since F (X,∆,γ) is max. destab.

Claim 8.8 thus follows.

Notation 8.9. Following Notation 2.35 the foliation F sat induces a decomposition ∆ = ∆ trans + ∆ ntrans .

Step 3: Construction of a morphism. We show that the foliation F sat is algebraic and therefore defines an essentially equidimensional rational map. The algebraicity criterion that we employ goes back to Hartshorne, [Har68, Thm. 6.7]. We refer the reader to [KST07] or to one of the papers [Bos01, BM01, BM16] for a thorough discussion.

Claim 8.10. There exists a normal, projective variety Z, and a dominant, essentially equidimensional, rational map ψ : X Z such that the sheaves T X/Z and F sat agree.

Proof of Claim 8.10. In the setting at hand, amplitude of the restricted foliation F | C implies that any leaf of F that intersects C is automatically algebraic; we refer the reader to [KST07, Thm. 1] for a convenient reference. Since C is general in a dominating family, this gives rise to a rational map φ : X X × Chow(X), x → x, leaf through x .

Since X is normal, it follows from Zariski's main theorem, [Har77, V Thm. 5.2], that there exists a big open set U ⊆ X where φ is well-defined. Observe that the following holds:

• The morphism φ| U has a right inverse and is therefore necessarily injective, in particular quasi-finite. • The image of the morphism φ is contained in the universal family that exists for the Chow variety. Let V be the normalisation of the closure of the image. The universal family over Chow(X) need not be normal. Normalising, we obtain a diagram as follows,

U

/ / well-defined, injective

' ' X φ / / _ _ _ V normalisation / / equidim.
, , Univ equidim.

/ / Chow(X).

More is true. Zariski's Main Theorem in the form of Grothendieck6 , asserts that the map U → V is in fact an open immersion. In summary, we see that the composed morphism U → Chow(X) is equidimensional. Let Z be the normalisation of the image, and ψ : X Z the induced map. The sheaves T X/Z and F agree on an open subset of U, where all spaces and foliations are regular, and all maps are well-defined and smooth. Since both are saturated subsheaves of T X , this suffices to show that they agree everywhere. Claim 8.10 follows.

Notation 8.11. Following Notation 2.21, the map ψ : X Z induces a decomposition ∆ = ∆ horiz + ∆ vert . Observation 8.12. The decomposition of ∆ agrees with that coming from the foliation. In other words, ∆ trans = ∆ horiz and ∆ ntrans = ∆ vert .

Step 4: End of proof. To end the proof, we need to show that the rational map ψ satisfies Inequality (8.1.2). We aim to apply Proposition 4.6. To this end, recall from our construction that F = F sat ∩ T X (-log⌊∆⌋) and F (X,∆,γ) = γ [ * ] F ∩ T (X,∆,γ) . Item (4.6.3) of Proposition 4.6 thus gives an equality of intersection numbers

C γ • [γ [ * ] T X/Z ] = C γ • [F (X,∆,γ) ]
>0 by Obs. 8.4

+ C γ • [γ * ∆ trans ] + C γ • [effective] ≥0 since C γ movable > C γ • [γ * ∆ trans ].
In summary, we see that

C γ • [γ [ * ] T X/Z ] > C γ • [γ * ∆ trans ] and hence C • [T X/Z ] > C • [∆ trans
] as claimed. This finishes the proof of Theorem 8.1.

PROOF OF THE SEMIPOSITIVITY RESULT

We prove Theorem 5.3 in this section. With the preparations at hand, the proof is now quite short. We argue by contradiction and assume that Ω

[1] (X,∆,γ) is not γ-generically semipositive. As we have seen in Theorem 8.1, this implies the existence of a normal variety Z, a dominant, essentially equidimensional, rational map f : X Z, and a family (C t ) t∈T of curves that dominates X and avoids small sets, such that the following inequality holds for all t ∈ T,

[T X/Z ] • [C t ] > [∆ horiz ] • [C t ].
Recalling the description of T X/Z given in Lemma 2.31, this is equivalent to

[K X/Z + ∆ horiz -Ramification f ] • [C t ] < 0,
contradicting the positivity of relative dualising sheaves that was established in Theorem 7.1, and ending the proof of Theorem 5.3.

1. 3 .

 3 A note on further results. Constructing degenerate Kähler-Einstein metrics, Campana and Pȃun have established a second proof of Theorem 5.3 that works for Kähler manifolds, [CP14].

  (3.11.3) The three terms of (3.11.1) are equal away from supp Ramification( f ), and near general points of supp ∆ γ . (3.11.4) If the covering morphism γ is Galois, say with group G, then all sheaves appearing in (3.11.1) carry natural structures of G-sheaves, and the inclusions are inclusions of G-sheaves. (3.11.5) We have an equality of sheaves, det Ω

  pair, the notion of "adapted differentials" agrees with the notion introduced in earlier papers of Campana et al., cf. [JK11a, Sect. 2.D] and Campana's work referenced there.

Remark 4. 2 (

 2 Inclusions between tangent sheaves). Dualising, Remark 3.11 yields inclusions (4.2.1)

  (4.5.2) The cover γ is good. (4.5.3) The saturated subsheaf F ⊆ T X (-log⌊∆⌋) is actually a subbundle. (4.5.4) The subsheafG Y ⊆ T Ylog ∆ γ is a subbundle. Ditto for G Y ⊆ T (X,∆,γ)and G ⊆ γ [ * ] T X (-log⌊∆⌋). (4.5.5) If D ⊆ supp(∆ + Branch γ) is any component, then either F is everywhere transversal to D, or everywhere tangent to D.

  (6.4.1) The variety X m is smooth. (6.4.2) The µ m -exceptional set is of pure codimension one in X m . Let E m denote the associated reduced divisor. (6.4.3) The divisor D m + B Dm + E m has simple normal crossing support, where D m and B Dm denote the strict transforms D and B D , respectively.

  ) and as vol( A m ) = vol(A), we have volK X + D + 1 m B D ≥ c -n • vol(A). Theorem 6.1 now follows by taking m → ∞.Part II. Proof of the semipositivity result

  cover for (Z, OrbiBranch( f ))

  (8.3.1) The curve C γ is entirely contained in the smooth locus of Y and is itself smooth. (8.3.2) The sheaves F (X,∆,γ) | C γ and F sat (X,∆,γ) | C γ are locally free. Observation 8.4 (Positivity along C γ ). The locally free sheaf F (X,∆,γ) | C γ is semistable, of positive degree and therefore ample. The larger sheaf F sat (X,∆,γ) | C γ is likewise ample.

.

  Observation 8.7 (Regularity and amplitude along C). Since γ is finite, the curve C := γ(C γ ) is again a general member in a dominating family of curves that avoids small sets. It follows that C is contained in the smooth locus of X and that F is locally free near C. In particular,F sat (X,∆,γ) | C γ = (γ| C γ ) * (F | C ),and [Laz04b, Prop. 6.1.8] therefore implies that F | C and F sat | C are both ample.

  17. In the setting of Definition 2.16, observe thatf supp Ramification f ⊆ supp Branch f . Construction 2.18 (Local normal form). Let f : X → Z be an equidimensional morphism of normal varieties, of relative dimension d. Let Z • ⊆ Z be the largest open set such that both Z • and Z • ∩ (supp Branch f ) are smooth. Let X • ⊆ f -1 (Z • ) bethe largest open set such that both X • and X • ∩ (supp Ramification f ) are smooth, and such that the following restrictions of f are smooth morphisms,

	If f is proper, then equality holds.
	2.3.3. Local normal form. In this section we give explicit description of equidimen-
	sional morphisms in a suitably chosen local analytic coordinate system.

  the fibre over z. We say that f has connected fibres if f Defn has connected fibres. If ∆ is any prime divisor on X, observe that ∆ intersects Defn( f ) non-trivially. Call ∆ horizontal if ∆ ∩ Defn( f ) dominates Z, otherwise call it vertical. If ∆ is any Q-Weil divisor, there exists an associated decomposition ∆ = ∆ horiz + ∆ vert .

	Notation 2.21 (Horizontal and vertical divisors). Let f : X	Z be a rational map
	between normal varieties. Definition 2.22. Let f : X	Z be a rational map between normal varieties. We say
	that f is essentially equidimensional if there exists an open set U ⊆ Defn( f ) that is
	big in X such that f | U is an equidimensional morphism.
	Remark 2.23. Recall from Definition 2.10 that equidimensional morphisms (and
	hence essentially equidimensional maps) are dominant.
	Definition 2.24 (Ramification and branch divisors for essentially equidimensional
	maps). Let f : X	Z be an essentially equidimensional rational map between normal
	varieties as in Definition 2.22. Define
		Ramification

  7.1) are equal away from supp Ramification( f ), and near supp ∆ γ . (3.7.4) If the covering morphism γ is Galois, say with group G, then all sheaves appearing in (3.7.1) carry natural structures of G-sheaves, and the inclusions are inclusions of G-sheaves. Remark 3.8 (Local description of adapted differentials). The inclusions (3.7.1) can be written down in local coordinates, near any given point y ∈ Y. If γ is étale at y, or if y ∈ supp ∆ γ then all three sheaves agree, and there is nothing much to do. Let us therefore assume that y ∈ Ramification( f ) \ supp ∆ γ . Choose local coordinates as in Setup 3.1 and follow the notation introduced there.

Near y the sheaves Ω 1 Y (log ∆ γ ) and Ω 1 Y agree, and so do γ * Ω 1 X (log⌊∆⌋) and γ * Ω 1 X . The sheaf Ω 1 Y is freely generated as an O Y -module by symbols dy 0 , . . . , dy n . The sheaves of (3.7.1) are then generated as follows,

  • ⊆ X and Y • ⊆ Y be the maximal open sets where f can locally be written in normal form and where supp(∆ + Branch γ) and supp(γ * ∆ + Ramification γ) are both smooth; these are big open subsets of X and Y, respectively. Let ι : Y • → Y be the inclusion map, and set ∆

  positive rank. The twisted weak positivity theorem of [Fuj14, Thm. 1.1] asserts that E is weakly positive. In other words, cf. [Fuj14, Sect. 7], there exists a dense, Zariski-open set Y • ⊆ Y such that E is weakly positive over U, in the sense of Viehweg, [Vie95, Defn. 2.11]. Its pull-back f * E is then likewise weakly positive, [Vie95, Lem. 2.15], and so is the target of the natural, non-trivial morphism

See[START_REF] Grothendieck | Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas[END_REF] Zariski's main theorem in the form ofGrothendieck and [GKP16, Thm.

3.8] for the precise statement used here. A full proof is found in the extended version of[START_REF] Greb | Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties[END_REF], available on the arXiv.

As introduced in Notation 2.34.

See Definition 2.22 on page 7.

See Notation 2.2 on page 4.

Zariski's Main Theorem in the form of Grothendieck is found in[START_REF] Grothendieck | Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas[END_REF]. We refer the reader to[START_REF] Greb | Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties[END_REF] Sect. 3.4] for a more detailed discussion.

Stefan Kebekus gratefully acknowledges support through a joint fellowship of the Freiburg Institute of Advanced Studies (FRIAS) and the University of Strasbourg Institute for Advanced Study (USIAS). Behrouz Taji was partially supported by the DFG-Graduiertenkolleg GK1821 "Cohomological Methods in Geometry" at Freiburg.