Serge Cantat

Romain Dujardin

FINITE ORBITS FOR LARGE GROUPS OF AUTOMORPHISMS OF PROJECTIVE SURFACES

We study finite orbits for non-elementary groups of automorphisms of compact projective surfaces. In particular we prove that if the surface and the group are defined over a number field k and the group contains parabolic elements, then the set of finite orbits is not Zariski dense, except in certain very rigid situations, known as Kummer examples. Related results are also established when k " C. An application is given to the description of "canonical vector heights" associated to such automorphism groups.

CONTENTS

1. Introduction 1 2. Very general Wehler surfaces 6 3. Non-elementary groups and invariant curves 13 4. Complex tori and Kummer examples 22 5. Unlikely intersections for non-elementary groups 27 6. Around Theorem B: consequences and comments 43 7. From number fields to C 46 8. Canonical vector heights 52 Appendix A. Kummer surfaces of type (6) 64 References 64

INTRODUCTION

1.1. Setting. Let X be a complex projective surface, and denote by AutpXq its group of automorphisms. The group AutpXq acts on the Néron-Severi group NSpX; Zq (resp. on the cohomology group H 2 pX; Zq); this gives a linear representation

f Þ Ñ f ˚(1.1)
Date: December 11, 2020. from AutpXq to GLpNSpX; Zqq (resp. GLpH 2 pX; Zqq). By definition, a subgroup Γ of AutpXq is non-elementary if its image Γ ˚Ă GLpNSpX; Zqq (resp. Ă GLpH 2 pX; Zqq) contains a free group of rank ě 2; equivalently, Γ ˚does not contain any abelian subgroup of finite index (see [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] for details and examples).

Our purpose is to study the existence and abundance of finite (or "periodic") orbits under such non-elementary group actions. Several possible scenarios can be imagined:

(a) a large -that is Zariski dense or dense-set of finite orbits; (b) finitely many finite orbits; (c) no finite orbit at all.

For a cyclic group generated by a single automorphism, the situation is well understood: in many cases the set of periodic points is large (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF] for an introduction to this topic, and [START_REF] Xie | Periodic points of birational transformations on projective surfaces[END_REF] for the case of birational transformations). On the other hand, for non-elementary groups, we expect the existence of a dense set of periodic points to be a rare phenomenon; this expectation will be confirmed by our results.

In fact, the only examples we know for situation (a) are given by abelian surfaces and their siblings, Kummer surfaces. Here, by Kummer surface we mean a smooth surface X which is a (non necessarily minimal) desingularization of the quotient A{G of an abelian surface A " C 2 {Λ by a finite group G Ă AutpAq. For instance, if G is generated by the involution px, yq Þ Ñ p´x, ´yq on A, we find the so-called classical Kummer surfaces and their blow-ups (see [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]). Given a subgroup Γ Ă AutpXq, we say that the pair pX, Γq is a Kummer group if X is a Kummer surface and Γ comes from a subgroup of AutpAq which normalizes G; precise definitions are given in §5.7. If Γ is a group of automorphisms of an abelian surface A fixing the origin 0 P A, then all torsion points are Γ-periodic. This implies that most Kummer groups have a dense set of finite orbits (see Proposition 4.5).

1.2. Main results. We first illustrate property (c) in the family of Wehler surfaces that is, smooth surfaces X Ă P 1 ˆP1 ˆP1 defined by a polynomial equation of degree p2, 2, 2q. Such an X is a K3 surface and generically its automorphism group is generated by three involutions, each of them swapping one coordinate on X. We focus on these examples because they occupy a central position in the dynamical study of surface automorphisms, both from the ergodic and arithmetic points of view (see e.g. [START_REF] Joseph | Rational points on K3 surfaces: a new canonical height[END_REF][START_REF] Kawaguchi | Canonical heights, invariant currents, and dynamical eigensystems of morphisms for line bundles[END_REF][START_REF] Curtis | Dynamics on K3 surfaces: Salem numbers and Siegel disks[END_REF]).

Theorem A. For a very general Wehler surface X, every orbit under AutpXq is Zariski dense. In particular there is no finite orbit under the action of AutpXq.

Unfortunately, from the nature of its proof, this theorem has an obvious limitation: it does not allow to single out any explicit example satisfying Property (c).

Our main result concerns Property (b). To state it, recall that there are three types of automorphisms, characterized by the behavior of the linear endomorphism f ˚(see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]).

If f ˚has finite order, then f is elliptic. Otherwise, f is either parabolic or loxodromic: it is parabolic if f ˚has infinite order, but none of its eigenvalues has modulus ą 1; it is loxodromic if some eigenvalue λpf q of f ˚has modulus |λpf q| ą 1 (in that case λpf q is unique and λpf q P p1, `8q). A non-elementary group of automorphisms contains a non-abelian free group all of whose non-trivial elements are loxodromic, and a group containing both loxodromic and parabolic elements is automatically non-elementary.

Theorem B. Let X be a smooth projective surface, defined over some number field k. Let Γ be a subgroup of AutpXq, also defined over k, containing both parabolic and loxodromic automorphisms. If the set of finite orbits of Γ is Zariski dense in X, then pX, Γq is a Kummer group.

When Γ is non-elementary there is a maximal Γ-invariant curve D Γ ; more precisely, either Γ does not preserve any curve, or there exists a unique, maximal, Γ-invariant Zariski closed subset of pure dimension 1. This curve D Γ can be contracted to yield a (singular) complex analytic surface X 0 and a Γ-equivariant birational morphism π 0 : X Ñ X 0 .

(1.2)

Thus if pX, Γq is not a Kummer group, property (b) holds on X 0 . It turns out that when Γ contains a parabolic automorphism, X 0 is projective (see Proposition 3.9). Another result, which plays an important role in the proof of Theorem B is the following Theorem D: any non-elementary subgroup Γ Ă AutpXq contains a loxodromic element whose maximal periodic curve is equal to D Γ (see Section 3 for the precise statement).

Let us stress that even if X and Γ are defined over k, Theorem B concerns orbits of Γ in XpCq. In this respect it is very different in spirit from the results of [START_REF] Joseph | Rational points on K3 surfaces: a new canonical height[END_REF] or [START_REF] Kawaguchi | Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint[END_REF], in which finiteness results are obtained for the number of periodic orbits of elementary groups acting on Xpk 1 q, where k 1 is a fixed finite extension of k, which ultimately rely on Northcott's theorem.

Under the assumptions of Theorem B, we obtain the following corollaries (see Corollaries 6.1, and 6.2 and Proposition 4.5 for details):

-If Γ does not preserve any algebraic curve and X is not an abelian surface, then Γ admits at most finitely many finite orbits. -If C is an irreducible curve containing infinitely many Γ-periodic points, then either C is Γ-periodic or pX, Γq is a Kummer group and C comes from a translate of an abelian subvariety. In particular if C has genus ě 2, it contains at most finitely many Γ-periodic points.

-If Γ has a Zariski dense set of finite orbits, then its finite orbits are dense in XpCq for the Euclidean topology; furthermore if f 1 and f 2 are two loxodromic automorphisms in Γ, their periodic points coincide, except for at most finitely many of them which are located on Γ-invariant curves.

As we shall see in Remark 6.6, the last statement provides a partial answer to a question of Kawaguchi. 1.3. Proof strategy and extension to complex coefficients. Let us say a few words about the proof of Theorem B (a more detailed outline is given in §5.1). Given two "typical" loxodromic elements f, g in Γ, intuition suggests that Perpf q X Perpgq cannot be Zariski dense unless some "special" phenomenon happens. This situation has been referred to as an unlikely intersection problem in the algebraic dynamics literature. Previous work on this topic suggests to handle this problem using methods from arithmetic geometry (see e.g. [START_REF] Baker | Preperiodic points and unlikely intersections[END_REF][START_REF] Dujardin | The dynamical Manin-Mumford problem for plane polynomial automorphisms[END_REF]). In this respect a key idea would be to use arithmetic equidistribution (see [START_REF] Yuan | Big line bundles over arithmetic varieties[END_REF][START_REF] Berman | Growth of balls of holomorphic sections and energy at equilibrium[END_REF]) to derive an equality µ f " µ g between the measures of maximal entropy of f and g. Unfortunately we do not know how to infer rigidity results directly from this equality, so the proof of Theorem B is not based on this sole argument. To reach a concluson, we make use of the dynamics of the whole group Γ, in particular of the classification of Γ-invariant measures (see [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Cantat | Dynamics of automorphism groups of complex projective surfaces: parabolic maps and invariant measures[END_REF]), together with the classification of loxodromic automorphisms f whose measure of maximal entropy µ f is absolutely continuous with respect to the Lebesgue measure (see [START_REF] Cantat | Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy[END_REF][START_REF] Filip | Kummer rigidity for K3 surface automorphisms via Ricci flat metrics[END_REF]). The existence of parabolic elements in Γ is required at three important stages, including the arithmetic step; in particular we are not able to prove Theorem B without assuming that Γ contains parabolic elements (see §6.3 for a more precise discussion).

Even if arithmetic methods lie at the core of the proof of Theorem B, it is natural to expect that the assumption that X and Γ be defined over a number field is superfluous. We are indeed able to get rid of it when Γ has no invariant curve.

Theorem C. Let X be a compact Kähler surface which is not a torus. Let Γ be a subgroup of AutpXq which contains a parabolic element and does not preserve any algebraic curve. Then Γ admits only finitely many periodic points.

The proof of Theorem C is based on specialization arguments, inspired notably by the approach of [START_REF] Dujardin | The dynamical Manin-Mumford problem for plane polynomial automorphisms[END_REF] (see Section 7). It applies, for instance, to the action of Γ " AutpXq on any unnodal Enriques surface X, and to the foldings of euclidean pentagons with generic side lengths (see [18, §3] for details on these examples).

We conclude this introduction by explaining two further applications of Theorem B.

1.4. Canonical vector heights. Theorem B will be applied to answer a question of Baragar on the existence of certain canonical heights (see [START_REF] Baragar | Canonical vector heights on K3 surfaces with Picard number three-an argument for nonexistence[END_REF][START_REF] Baragar | K3 surfaces with Picard number three and canonical vector heights[END_REF][START_REF] Kawaguchi | Canonical vector heights on K3 surfaces-a nonexistence result[END_REF]).

Let X be a projective surface, defined over a number field k. Denote by PicpXq the Picard group of X Q . The Weil height machine provides, for every line bundle L on X, a height function h L : XpQq Ñ R, defined up to a bounded error Op1q. This construction is additive, h aL`bL 1 " ah L `bh L 1 `Op1q for all pairs pL, L 1 q P PicpXq 2 and all coefficients pa, bq P Z 2 . When L " O X p1q for some embedding X Ă P N k , then h L coincides with the usual logarithmic Weil height.

If f is a regular endomorphism of X defined over k and L is an ample line bundle such that f ˚L " L bd for some integer d ą 1, then h L ˝f " dh L `Op1q. Tate's renormalization trick ĥL pxq :" lim nÑ`8

1 d n h L pf n pxqq (1.3)
provides a canonical height for f and L that is, a function ĥL : XpQq Ñ R `such that ĥL " h L `Op1q and ĥL ˝f " d ĥL exactly, with no error term. This construction was extended to loxodromic automorphisms of projective surfaces by Silverman, Call, and Kawaguchi (see [START_REF] Joseph | Rational points on K3 surfaces: a new canonical height[END_REF][START_REF] Call | Canonical heights on varieties with morphisms[END_REF][START_REF] Kawaguchi | Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint[END_REF]): in this case one obtains a pair of canonical heights ĥf satisfying ĥf ˝f ˘1 " λpf q ˘ĥ f . (Note that here ĥf and ĥf are Weil heights associated to R-divisors.) If Γ is an infinite subgroup of AutpXq, also defined over k, it is natural to ask whether a Γ-equivariant family of heights can be constructed. Specifically, one looks for a family of representatives ĥL of the Weil height functions, i.e. ĥL " h L `Op1q for every L in PicpX; Rq :" PicpXq b Z R, depending linearly on L, and satisfying the exact relation ĥL pf pxqq " ĥf ˚Lpxq p@x P XpQqq (1.4) for every pair pf, Lq P Γ ˆPicpX; Rq (see §8 for details). A prototypical example is given by the Néron-Tate height, when Γ is the group of automorphisms of an abelian surface preserving the origin. Such objects were named canonical vector heights ( 1 ) by Baragar in [START_REF] Baragar | Canonical vector heights on algebraic K3 surfaces with Picard number two[END_REF]. He proved their existence when X is a K3 surface with Picard number 2, in which case AutpXq is virtually cyclic. He also gave evidence for their non-existence on certain Wehler surfaces (see [START_REF] Baragar | K3 surfaces with Picard number three and canonical vector heights[END_REF]). In [START_REF] Kawaguchi | Canonical vector heights on K3 surfaces-a nonexistence result[END_REF] Kawaguchi obtained a complete proof of this non-existence for an explicit family of Wehler surfaces; his argument relies on the study of Γ-periodic orbits. Extending Kawaguchi's methods and using Theorem B, we completely solve the existence problem of canonical vector heights for non-elementary groups with parabolic elements: let X be a smooth projective surface and Γ be a non-elementary subgroup of AutpXq containing parabolic elements, both defined over a number field k; if pX, Γq possess a canonical vector height, then X is an abelian surface and Γ has a finite orbit (see Theorem E in Section 8). The second assertion implies that, after conjugation by a translation, a finite index subgroup of Γ preserves the neutral element of the abelian surface X, in particular the Néron-Tate height provides a canonical vector height, and 1 The name "vector height" comes from the following viewpoint. Assume PicpX; Rq " NSpX; Rq, and fix a basis L i of PicpX; Rq. For a point x in XpQq, consider the vector ph Li pxqq P R ρ , where ρ " dim R PicpX; Rq. Then, the equivariance property (1.4) can be phrased in terms of this vector, hence the terminology.

we explain how all possible canonical vector heights are derived from the Néron-Tate height (see Theorems E' and E" below for precise statements). 1.5. Stationary measures. Another application, which was our primary source of motivation for this work, concerns the classification of invariant and stationary measures.

Assume that X and Γ are defined over R; in particular, Γ acts on the real part XpRq of X, which we assume here to be non-empty. The group Γ permutes the connected components of XpRq and, choosing one component XpRq i in each Γ-orbit, the surface XpRq splits into a finite, disjoint union of real analytic and Γ-invariant surfaces XpRq " ğ iPI ΓpXpRq i q.

(1.5)

For simplicity, as in Theorem C, suppose that X is not abelian, Γ contains both loxodromic and parabolic elements, and Γ does not preserve any algebraic curve D Ă X.

Let ν be a probability measure on AutpXq, whose support is a finite set generating Γ.

Using the results of [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Cantat | Dynamics of automorphism groups of complex projective surfaces: parabolic maps and invariant measures[END_REF][START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] we infer that: the set of ν-stationary probability measures on XpRq coincides with that of Γ-invariant probability measures and is a finite dimensional simplex, whose extremal points are given by:

finitely many real analytic area forms ω i , one for each orbit ΓpXpRq i q, with support equal to ΓpXpRq i q; -the uniform counting measures on the (finitely many) finite orbits of Γ.

Example. Suppose X is a real K3 surface, with XpRq ‰ H. Then, XpRq is orientable, and there is a non-vanishing section Ω of the canonical bundle K X which induces a positive area form Ω R on XpRq (see [START_REF] Cantat | Dynamics of automorphism groups of complex projective surfaces: parabolic maps and invariant measures[END_REF] for instance). The area forms mentioned in the first item are the restrictions of Ω R to the surfaces ΓpXpRq i q, up to some normalization factors. So, if X is a very general real Wehler surface with XpRq ‰ H, and if ν generates AutpXq, Theorem A implies that the only ν-stationary measures are convex combinations of restrictions of the natural area form Ω R to the components of XpRq (this result was announced in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]).

1.6. Organization of the paper. We start by proving Theorem A in Section 2, which is independent of the rest of the paper. In Section 3 we study invariant curves for loxodromic automorphisms and non-elementary groups. In particular we obtain an effective bound for the degree of a curve invariant under a loxodromic automorphism (see Proposition 3.7) and prove Theorem D. In Section 4 we briefly discuss the case of tori and review the Kummer construction. The core of the paper is Section 5, in which we develop the arithmetic method outlined above and establish Theorem B. Section 6 is devoted to consequences of Theorem B, and related comments. We prove Theorem C in Section 7. Finally, canonical vector heights are discussed in Section 8, where we solve Baragar's problem. Some open problems and possible extensions of our results are discussed in § §6.3 and 7.4.

Acknowledgement. We warmly thank Pascal Autissier, Antoine Chambert-Loir, Marc Hindry, Mattias Jonsson, and Junyi Xie for useful discussions or comments.

VERY GENERAL WEHLER SURFACES

Consider the family of Wehler surfaces described in Section 3.1 of [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] (or in [START_REF] Baragar | Canonical vector heights on K3 surfaces with Picard number three-an argument for nonexistence[END_REF][START_REF] Cantat | Dynamique des automorphismes des surfaces K3[END_REF][START_REF] Kawaguchi | Canonical vector heights on K3 surfaces-a nonexistence result[END_REF][START_REF] Curtis | Dynamics on K3 surfaces: Salem numbers and Siegel disks[END_REF]). In this section we prove Theorem A. Recall the statement: Theorem 2.1. If X Ă P 1 ˆP1 ˆP1 is a very general Wehler surface, then AutpXq does not preserve any non-empty, proper, and Zariski closed subset of X.

Here, by very general, we mean that this property holds in the complement of a set of countably many hypersurfaces in the space of surfaces of degree p2, 2, 2q in P 1 ˆP1 ˆP1 . The proof follows from an elementary but rather tedious parameter counting argument. As we shall see in §2.5, such a statement does not hold if we replace AutpXq by a thin non-elementary subgroup.

2.1. Notation and preliminaries. We use the notation of [18, §3.1]: M " P 1 ˆP1 ˆP1 , with affine coordinates px, y, zq (denoted px 1 , x 2 , x 3 q in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]), π 1 , π 2 , and π 3 are the projections on the first, second, and third factors, and π ij is the projection pπ i , π j q onto P 1 ˆP1 . Then L i " π i pOp1qq, L " L 2 1 bL 2 2 bL 2 3 , and X Ă M is a member of the linear system |L|. In the affine coordinates px, y, zq, X is defined by a polynomial equation of degree p2, 2, 2q, which we write P px, y, zq " A 222 x 2 y 2 z 2 `A221 x 2 y 2 z `¨¨¨`A 100 x `A010 y `A001 z `A000 . (2.1)

We thus see that H 0 pM, Lq is of dimension 27 and since the equation tP " 0u is defined up to multiplication by a complex scalar, the family of Wehler surfaces X is 26dimensional. Modulo the action of G " PGLp2, Cq 3 they form an irreducible family of dimension 17.

It was shown in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Prop. 3.1] that there exists a Zariski open set W 0 Ă |L| of surfaces X P |L| such that (i) X is a smooth K3 surface; (ii) each of the three projections pπ ij q X : X Ñ P 1 ˆP1 is a finite map, that is, X does not contain any fiber of π ij : M Ñ P 1 ˆP1 .

From now on, we suppose that X belongs toW 0 . Let i, j, k be three indices with ti, j, ku " t1, 2, 3u. Denote by σ i : X Ñ X the involutive automorphism of X that permutes the points in the fibers of the 2-to-1 branched covering pπ jk q X : X Ñ P 1 ˆP1 . By [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Lem. 3.2], the three involutions σ i generate a non-elementary subgroup of AutpXq. This subgroup is isomorphic to Z{2Z ‹ Z{2Z ‹ Z{2Z, it preserves the subspace of NSpX; Zq generated by the Chern classes of the L i , and its action on this subspace is given by the matrices in Equation (3.4) of [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]. Then f ij " σ i ˝σj is a parabolic automorphism of X, preserving the genus 1 fibration π k : X Ñ P 1 . Moreover, if X is very general the L i generate NSpX; Zq (see [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Prop. 3.3]).

2.2. Invariant curves.

Proposition 2.2. If X P W 0 , AutpXq does not preserve any algebraic curve.

This is a direct consequence of the considerations of the previous paragraph, together with the following more precise result.

Lemma 2.3. Let X be a smooth Wehler surface. Assume that the three involutions σ i induce a faithful action of the group Z{2Z ‹ Z{2Z ‹ Z{2Z. Then the group generated by the σ i does not preserve any curve.

Proof. Assume that C is an invariant curve. Since no curve can be contained simultaneously in fibers of π 1 , π 2 and π 3 , without loss of generality, we may suppose that π 1 : C Ñ P 1 pCq is dominant. Then the automorphism f 23 " σ 2 ˝σ3 has finite order: indeed, on a general fiber F of π 1 , it acts as a translation that preserves the non-empty finite set F XC. This contradicts the fact that f 23 is parabolic and finishes the proof. Thus, to prove Theorem 2.1, we are left to prove the non-existence of periodic orbits, which is the purpose of the following paragraphs.

Elliptic curves.

Here we study [START_REF] Baragar | Canonical vector heights on algebraic K3 surfaces with Picard number two[END_REF][START_REF] Baragar | Canonical vector heights on algebraic K3 surfaces with Picard number two[END_REF] curves in dimension 2. We keep notation as in §2.1. Let us consider the line bundles L i " π i pOp1qq on P 1 ˆP1 and set L " L 2 1 bL 2 2 . Fix (affine) coordinates px, yq on P 1 ˆP1 , with x and y in CYt8u. A curve C Ă P 1 ˆP1 in the linear system |L| is given by an equation of degree [START_REF] Baragar | Canonical vector heights on algebraic K3 surfaces with Picard number two[END_REF][START_REF] Baragar | Canonical vector heights on algebraic K3 surfaces with Picard number two[END_REF] in px, yq. Assume that C contains the points p0, 0q, p8, 0q, and p0, 8q and that it is smooth at the origin, with a tangent line given by x `y " 0. Then its equation reduces to the form αx 2 y 2 `βx 2 y `γxy 2 `δxy `εpx `yq " 0 (2.2)

for some complex numbers α, β, γ, δ, and ε, with ε ‰ 0. Denote this curve by C pα,β,γ,δ,εq . For a general choice of these parameters, C is a smooth curve of genus 1. We will need the following more precise result.

Lemma 2.4. Fix pβ, γ, δ, εq with ε ‰ 0. Then for general α, C pα,β,γ,δ,εq is smooth.

Proof. An easy explicit calculation shows that the points of C on t8uˆP 1 and P 1 ˆt8u are smooth unless α " β " γ " 0. So for α ‰ 0, C has no singular point at infinity. Now, viewing the equation (2.2) as a quadratic equation in x depending on the variable y, we can consider its discriminant ∆ x " ∆ x pyq, which is a polynomial of degree 4 in y that detects fibers C ˆtyu intersecting C at a single point (for those values y for which C X C ˆtyu is contained in C 2 , that is the polynomial in x is of degree 2). It is easy to check that if px, yq is a singular point of C, then y must be a multiple root of ∆ x . Hence if y Þ Ñ ∆ x pyq only has simple roots, C is smooth in C 2 . Thus it is enough to check that if pβ, γ, δ, εq is an arbitrary 4-tuple such that ε ‰ 0, ∆ x has only simple roots for general α.

A simple calculation shows that ∆ x pyq " ay 4 `by 3 `cy 2 `dy `e, where only b depends on α, with bpαq " 2γδ ´4αε, and e " ε 2 ‰ 0. Now the discriminant of ∆ x , as a degree 4 polynomial in y, is a polynomial expression in pa, b, c, d, eq, and as a polynomial in b it has a unique leading term 27b 4 e 2 . So, pβ, γ, δ, εq being fixed, with ε ‰ 0, this discriminant depends non-trivially on α; for a general α, this discriminant is not zero thus ∆ x has four distinct roots, so that C is smooth, as was to be proved.

As for Wehler surfaces, there are two involutions σ 1 and σ 2 on C, respectively permuting the points in the fibers of the projections pπ 2 q |C : C Ñ P 1 and pπ 1 q |C : C Ñ P 1 , that is, σ i changes the i-th coordinate, while keeping the other ones unchanged. The composition f " σ 1 ˝σ2 is a translation on C mapping p0, 8q to p8, 0q; in particular, f is not the identity. Lemma 2.5. Fix pβ, γ, δ, εq with ε ‰ 0 and assume that the curve C p0,β,γ,δ,εq is smooth. Then the dynamics of the translation f on C pα,β,γ,δ,εq varies non-trivially with α: it is periodic for a countable dense set of α's, and non-periodic for the other parameters.

Proof. For α in the complement of a finite set, C α :" C pα,β,γ,δ,εq is a smooth curve of genus 1, and f acts as a translation on C α . Let us analyze the orbit of p0, 8q. Denote by u, v P C Y t8u the complex numbers such that σ 2 p8, 0q " p8, vq and pu, vq " σ 1 p8, vq " f 2 p0, 8q. The translation f is periodic of period 2 if and only if pu, vq " p0, 8q, if and only if p8, 8q is a point of C α , if and only if α " 0. Hence, f is periodic of period 2 on C 0 but after perturbation it is not of period 2 anymore. For small α we can write C " C{Λ α for some lattice Λ α " Z `Zτ pC α q and f pzq " z `tpC α q, with tpC α q and τ pC α q depending holomorphically on the parameters α. If we further decompose tpC α q " apC α q `bpC α qτ pC α q, where a and b are two real analytic functions with values in R, then both a and b must be non constant. Indeed if one of them were constant, then the other one would be a non-constant real holomorphic function, which is impossible (see [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF]Prop. 2.2] for a similar argument). The result follows.

2.4. Proof of Theorem 2.1.

2.4.1.

From finite orbits to fixed points. Let us form the universal family X Ă W 0 ˆM , where W 0 Ă |L| is the open set defined in Section 2.1: the fiber of the projection X Ñ W 0 above X P W 0 is precisely the surface X Ă M .

The group Z{2Z ‹ Z{2Z ‹ Z{2Z acts by automorphisms on X , preserving each fiber of X Ñ W 0 : the generators of the first, second and third Z{2Z factors give rise to involutions r σ 1 , r σ 2 and r σ 3 which, when restricted to a fiber X, correspond to the automorphisms σ i P AutpXq. These involutions r σ i extend to birational involutions of the Zariski closure X Ă |L| ˆM .

Remark 2.6. If X P |L| is smooth and contains a fiber V " tpx 0 , y 0 qu ˆP1 Ă X of π 12 , the curve V is contained in the indeterminacy locus of σ3 (one may consult [START_REF] Cantat | Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups[END_REF] for further results: see Theorem 3.3 and the proof of its third and fourth assertions).

Consider the group Z{2Z ‹ Z{2Z ‹ Z{2Z acting on X . Its restriction to the fiber X gives a subgroup Γ of AutpXq. Let d be a positive integer. There are only finitely many homomorphisms from Z{2Z‹Z{2Z‹Z{2Z to groups of order ď d!, and the intersection of the kernels of these homomorphisms is a normal subgroup of finite index. Denote by Γ d the corresponding subgroup of AutpXq. If Γ has an orbit of cardinality ď d on some surface X, then this orbit is fixed pointwise by Γ d . Let us introduce the subvariety

Z d " tpX, xq ; x P X and @f P Γ d , f pxq " xu Ă X . (2.3)
Since X Ñ W 0 is proper, from this discussion we get:

Lemma 2.7. The following properties are equivalent:

(1) for a very general surface X P |L|, every orbit of Γ in X is infinite.

(2) for every d ě 1, the projection Z d Ñ W 0 is not surjective;

: X Þ Ñ s X of X Ñ W 0 such that s X is fixed by Γ d ; equivalently, the image of s is contained in Z d .
The group G " PGL 2 pCq ˆPGL 2 pCq ˆPGL 2 pCq acts on M and on |L|, preserving W 0 . Recall that modulo the action of this group, the space of Wehler surfaces is irreducible and of dimension 17.

Case 1.

Let us first assume that we can find U such that s X is fixed neither by σ1 , σ2 , nor σ3 . As in Lemma 2.4 this implies that for each pair of indices i ‰ j, the fiber C of pπ i q X : X Ñ P 1 through s X is smooth near s X and s X P C is not a ramification point of the projection pπ j q |C : C Ñ P 1 .

As in Section 2.1, fix coordinates px, y, zq on M " pP 1 q 3 with x, y, and z in CYt8u. Modulo the action of G, we may assume that for every X in U , (a) the point s X is the point p0, 0, 0q in pP 1 q 3 ; (b) X contains p8, 0, 0q, p0, 8, 0q, and p0, 0, 8q; (c) the tangent plane to X at the origin is given by the equation x `y `z " 0; (d) the coefficients of x 2 y 2 z 2 and x, y and z in the equation of X are all equal to the same complex number.

Note that (a) can be achieved by a single translation, (b) can be obtained by transformations of the form px, y, zq Þ Ñ p x x´α , y y´β , z z´γ q, (c) is achieved by the action of diagonal maps (note that by our assumption, the tangent plane to X at the origin s X " p0, 0, 0q cannot be one of the coordinate planes), and then we obtain (d) by the action of homotheties. After such a conjugation, the equation of X is of the form

Ax 2 y 2 z 2 `Bx 2 y 2 z `B1 x 2 yz 2 `B2 xy 2 z 2 `Cx 2 yz `C1 xy 2 z `C2 xyz 2 (2.4) `Dx 2 y 2 `D1 x 2 z 2 `D2 y 2 z 2 `Exyz `F x 2 y `F 1 x 2 z `F 2 xy 2 `F 3 y 2 z `F iv xz 2 `F v yz 2 `Gxy `G1 xz `G2 yz `Apx `y `zq " 0.
Since this equation is defined up to multiplication by an element of C ˚, we are left with 19 parameters. The automorphism f 12 " σ 1 ˝σ2 preserves the genus 1 fibration pπ 3 q |X : X Ñ P 1 . The fiber of pπ 3 q |X through p0, 0, 0q is a curve C Ă P 1 ˆP1 given by the equation

Dx 2 y 2 `F x 2 y `F 2 xy 2 `Gxy `Apx `yq " 0. (2.5)
Two cases need to be considered, depending on the smoothness of this curve.

-if this curve is singular, by Lemma 2.4 the coefficients in Equation (2.5) satisfy a non-trivial relation of the form P 3 pD, F, F 2 , G, Aq " 0;

-if it is smooth, consider an iterate f m 12 of f 12 in Γ d , with 1 ď m ď d!; then f m 12
is a translation of the genus 1 curve C that fixes s X , so that it fixes C pointwise. From Lemma 2.5, the coefficients in Equation (2.5) satisfy a relation of the form Q 3 pD, F, F 2 , G, Aq " 0.

In both cases we get a relation of the form R 3 pD, F, F 2 , G, Aq " 0 (with R 3 " P 3 or Q 3 ) that depends non-trivially on the first factor. Similarly, looking at the dynamics of f 23 " σ 2 ˝σ3 and f 31 " σ 3 ˝σ1 , we obtain two further relations of the form

R 1 pD 2 , F 2 , F v , G 2 , Aq " 0 and R 2 pD 1 , F 1 , F iv , G 1 , Aq " 0.
We claim that the subset defined by these 3 constraints is of codimension 3: indeed if we look at the subvariety cut out by the equations R i " 0, i " 1, 2, 3 and slice it by a 3-plane corresponding to the coordinates D, D 1 and D 2 , then by Lemmas 2.4 and 2.5 and the independence of variables, this slice is reduced to a point. This shows that the image of the section X Þ Ñ s X is at most 16-dimensional, which contradicts the fact that W 0 {G is of pure dimension 17. Thus our hypothesis on Z d cannot be true and Case 1 does not hold.

2.4.4. Case 2. If Case 1 does not hold, every point pX, px, y, zqq of Z d has the property: px, y, zq P X is a ramification point for at least one of the three projections pπ i q |X . Equivalently, every point of the finite orbit F " Γ d ps X q Ă X is fixed by at least one of the three involutions σ i . This case is simpler, since a direct count of parameters will lead to a contradiction.

' If a point of F were a ramification point of each pπ i q |X , this point would be a singularity of X, and X would not be in W 0 . So, each point of F is a ramification point for at least one and at most 2 of the projections.

' Now, assume that every point of F is a ramification point for exactly 2 of the projections. Choose a local section s X of Z d above a small open set U Ă W 0 (for the Euclidean topology), as in §2.4.3. Permuting the coordinates and using a translation in G, we assume that s X " p0, 0, 0q and s X is fixed by σ 2 and σ 3 . After this normalization, with notation as in Equation (2.1), we have A 010 " A 001 " A 000 " 0. Let s 1 X " σ 1 ps X q; this point is not equal to s X because otherwise X would be singular at s X . So, we may use a transformation of the form x Þ Ñ x x´α in G to assume that s 1 X " p8, 0, 0q (i.e. A 200 " 0). Now by our assumption, this second point must be fixed by σ 2 and σ 3 , which imposes two more constraints (A 201 " A 210 " 0). Now, consider the curve C 1 Ă X defined by the equation x " 0. Using elements of G acting on y and z by y Þ Ñ y y´β and z Þ Ñ z z´γ , we may assume that p0, 8, 8q is on C 1 and is a ramification point for pπ 2 q |C 1 . With such a choice, the coefficients of y 2 z 2 and y 2 z vanish. At this stage we did not use the diagonal action of pC ˚q3 , which stabilizes p0, 0, 0q, p8, 0, 0q, and p0, 8, 8q. With this we can impose for instance the same non-zero coefficients for the terms xy, yz, and zx, so we end up with 17 coefficients, hence at most 16 free parameters. Again this contradicts the fact that dimpW 0 q " 17.

' Now, assume that one of the points of the finite orbit F is fixed by σ 3 but not by σ 1 and σ 2 . The analysis is similar to that of the previous case. We may choose this point to be s X , and using the group G, we can arrange that s X " p0, 0, 0q, σ 1 ps X q " p8, 0, 0q, and σ 2 ps X q " p0, 8, 0q; with the notation from Equation (2.1), this means A 000 " A 200 " A 020 " 0. In addition A 001 " 0 because p0, 0, 0q is fixed by σ 3 . By our hypothesis, p8, 0, 0q is fixed by σ 2 or σ 3 (or both). This implies that at least one of A 210 or A 201 vanishes. Likewise A 120 A 021 " 0. Now consider the curve C 2 Ă X given by y " 0. Given the constraints already listed, the equation of C 2 can be written as αx 2 z 2 `βx 2 z `γxz 2 `δxz `εz 2 `ιx " 0.

(

2.6)

There are 4 ramification points for pπ 1 q |C 2 , counting with multiplicities, and none of them satisifies z " 0. So using z Þ Ñ z z´γ and x Ñ λx we may put one of them at p1, 0, 8q. This imposes α `γ `ε " 0 and β `δ " 0.

Finally, we may still use the subgroup tIduˆC ˚ˆC ˚Ă G, which fixes the four points p0, 0, 0q, p8, 0, 0q, p0, 8, 0q and p1, 0, 8q, to assume that the non-zero coefficients in front of yz, xz, and z 2 are equal. In conclusion, under our assumption we have found at least 10 independent linear constraints on the coefficients of the Wehler surface so again at most 16 free parameters remain.

So, in all cases we get a contradiction, and the proof of Theorem 2.1 is complete. , for some large positive integer m (as above, f 23 " σ 2 ˝σ3 , f 31 " σ 3 ˝σ1 ). The automorphism f 23 preserves the fibers of the projection pπ 1 q |X and its periodic points form a dense set of fibers (see [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Cantat | Dynamics of automorphism groups of complex projective surfaces: parabolic maps and invariant measures[END_REF] or §3.1.1 below). The intersection number between a fiber of pπ 1 q |X and a fiber of pπ 2 q |X is equal to 2. So, if m is big enough, f m 23 and f m 31 share a common fixed point (in fact » cm 4 common fixed points, for some c ą 0 as m goes to `8). If X P W 0 , xf m 23 , f m 31 y is non-elementary because the class c 1 P NSpX; Zq of the invariant fibration of f 23 is not fixed by f 31 , and vice-versa (see also Lemma 3.13 below). Taking a surface X P W 0 that is defined over Q, we get in particular: Proposition 2.8. For every integer N ě 0, there is a smooth Wehler surface X defined over Q and a non-elementary subgroup Γ of AutpX Q q with at least N fixed points. Remark 2.9. If X P W 0 and m ě 1, the group xf m 23 , f m 31 y has infinite index in AutpXq. Indeed, the index of xpσ 2 ˝σ3 q m , pσ 3 ˝σ1 q m y in Z{2Z ‹ Z{2Z ‹ Z{2Z is infinite. 2.5.2. Let us construct smooth Wehler surfaces for which the subgroup Γ of AutpXq generated by the three involutions σ i has a finite orbit. Using affine coordinates px, y, zq for pP 1 q 3 , set V " tpε 1 , ε 2 , ε 3 q ; ε i P t0, 8u @i " 1, 2, 3u. Observe that if V is contained in a Wehler surface X, then V is an orbit of Γ of size 8. Now, writing the equation of X as in (2.1), V Ă X if and only if A ijk " 0 for all triples pi, j, kq P t0, 2u 3 . The corresponding linear system has no base points, except from the points of V . If pA 100 , A 010 , A 001 q ‰ p0, 0, 0q, the surface is smooth at the base point p0, 0, 0q. The smoothness at the other 7 points of V is determined by similar constraints on the coefficients A ijk : for instance the smoothness at the point p8, 0, 0q is equivalent to pA 210 , A 201 , A 100 q ‰ p0, 0, 0q. Thus, the theorem of Bertini shows that a general member of the family of Wehler surfaces containing V is smooth, and the result follows.

NON-ELEMENTARY GROUPS AND INVARIANT CURVES

The main purpose of this section is to establish the following: Theorem D. Let X be a compact Kähler surface and let Γ be a subgroup of AutpXq containing a loxodromic element. Then there exists a loxodromic element f in Γ such that every f -periodic curve is Γ-periodic.

Along the way, some results of independent interest will be obtained: Proposition 3.7, which will be used in §7, gives an effective bound for the degree of a periodic curve under a loxodromic automorphism; Proposition 3.9 provides a singular model of pX, Γq without Γ-periodic curves, and discusses ampleness properties of some line bundles: this will be crucial for the study the dynamical heights in §5.

3.1. Preliminaries. Let X be a compact Kähler surface. By the Hodge index theorem, the intersection form x¨|¨y is non-degenerate and of signature p1, h 1,1 pXq ´1q on H 1,1 pX; Rq. Fix a Kähler form κ 0 on X, with ş X κ 0 ^κ0 " 1, denote its class by rκ 0 s, and define the positive cone in H 1,1 pX; Rq to be the set PospXq " tu P H 1,1 pX; Rq ; xu|uy ą 0 and xrκ 0 s|uy ą 0u.

(3.1)

Equivalently, PospXq is the connected component of tu P H 1,1 pX; Rq ; xu|uy ą 0u containing Kähler forms; in particular, its definition does not depend on κ 0 . This cone PospXq contains one of the two connected components, denoted H X , of the hyperboloid tu P H 1,1 pX; Rq ; xu|uy " 1u; we can identify H X with its projection PpH X q in the projective space PpH 1,1 pX; Rqq, and in doing so we get H X » PpH X q " PpPospXqq. Via this identification, the Hilbert metric on H X coincides with the hyperbolic metric induced by the intersection form (see [18, §2]), and the boundary BH X is identified to the projection of the isotropic cone in PpH 1,1 pX; Rqq. An automorphism of X has a type (elliptic, parabolic or loxodromic) according to the type of its induced action on H X . Given a subgroup Γ ď AutpXq, we denote by Γ par (resp. Γ lox ) the set of parabolic (resp. loxodromic) automorphisms in Γ.

3.1.1. Parabolic automorphisms (see [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF][START_REF] Cantat | Dynamics of automorphism groups of complex projective surfaces: parabolic maps and invariant measures[END_REF]). If g is parabolic, it permutes the fibers of a genus 1 fibration π g : X Ñ B g , and induces an automorphism g of the curve B g . The induced automorphism g has finite order, except maybe when X is a torus C 2 {Λ (see [START_REF] Cantat | Symétries birationnelles des surfaces feuilletées[END_REF]Prop. 3.6]).

If g is the identity, then g preserves each fiber of π g , acting as a translation on each smooth fiber. If U 0 is a disk in B g that does not contain any critical value of π g , the universal cover of π ´1 g pU 0 q is holomorphically equivalent to U 0 ˆC, with its fundamental group Z 2 acting by px, yq P U 0 ˆC Þ Ñ px, y `a `bτ pxqq for every pa, bq P Z 2 , where τ : U 0 Ñ C is a holomorphic function taking its values in the upper half plane. In these coordinates g lifts to a diffeomorphism gpx, yq " px, y `tpxqq for some holomorphic function t : U 0 Ñ C. The m-th iterate g m fixes pointwise a fiber txu ˆC{pZ ' Zτ pxqq if and only if mtpxq P Z ' Zτ pxq. The union of such fibers, for all m ě 1, form a dense subset of X. This comes from the fact that "t varies independently from τ ", a property which implies also that the differential of g m at a fixed point is, except for finitely many fibers, a 2 ˆ2 upper triangular matrix with 1's on the diagonal and a non-trivial lower left coefficient. We refer to [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF][START_REF] Cantat | Dynamics of automorphism groups of complex projective surfaces: parabolic maps and invariant measures[END_REF], and to the proof of Theorem 5.12 for a slightly different viewpoint on this property, using real-analytic coordinates.

The induced action g ˚on H 1,1 pX; Rq admits a simple description: if F is any fiber of π g , its class rF s P H 1,1 pX; Rq is fixed by g ˚, the ray R `rF s is contained in the isotropic cone, and 1 n 2 pg n q ˚w converges towards a positive multiple of rF s for every w P PospXq. In particular the class rF s is nef. Regarding the induced action on H X , PprF sq is the unique fixed point of the parabolic map g ˚on H X Y BH X (see [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF][START_REF] Cantat | Dynamics of automorphism groups of complex projective surfaces: parabolic maps and invariant measures[END_REF]).

Recall that a linear endomorphism of a vector space is unipotent if all its eigenvalues α P C are equal to 1; it is virtually unipotent if some of is positive iterates is unipotent. Since the topological entropy of g vanishes, Lemma 2.6 of [START_REF] Cantat | Automorphisms of compact Kähler manifolds with slow dynamics[END_REF] gives the following( 2 ). Lemma 3.1. Let X be a complex projective surface. If g P AutpXq is parabolic, then g ˚is virtually unipotent, both on NSpX; Rq and on H 2 pX; Rq. [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]). The dynamics of a loxodromic automorphism f is much richer. The isolated periodic points of f of period m equidistribute towards a probability measure µ f as m goes to `8, the topological entropy of f is positive, and µ f is the unique ergodic, f -invariant probability measure of maximal entropy.

Loxodromic automorphisms (see

We denote by λpf q the spectral radius of the induced automorphism f ˚on H 1,1 pXq, which is larger than 1. Then λpf q and 1{λpf q are eigenvalues of f ˚with multiplicity 1, with respective nef eigenvectors θ f and θ f which are isotropic and generate an f ˚invariant plane Π f Ă H 1,1 pX; Rq. Their projectivizations are the two fixed points on BH X of the induced loxodromic isometry of H X . The remaining eigenvalues have modulus 1. We normalize the eigenvectors θ f by imposing

xθ f |rκ 0 sy " xθ f |rκ 0 sy " 1 (3.2)
where κ 0 is the Kähler form introduced at the beginning of §3.1 (recall that xrκ 0 s|rκ 0 sy " 1). We set m f " 1 2 pθ f `θf q. With such a choice, xm f |m f y " 1 2 xθ f |θ f y ą 0.

Remark 3.2. Denote by Ang κ 0 pθ f , θ f q the visual angle between the boundary points Ppθ f q and Ppθ f q, as seen from rκ 0 s (or Pprκ 0 sq). Then

xm f |m f y " ˆsin ˆ1 2 Ang κ 0 pθ f , θ f q ˙˙2 " 1 2 `1 ´cospAng κ 0 pθ f , θ f qq ˘, (3.3) 2
This can also be proved directly. For instance, on NSpX; Rq this follows from the fact that the intersection form is negative definite on rF s K {RrF s and the lattice NSp; Zq is g ˚-invariant.

m f θ f θ f rκ 0 s FIGURE 1.
On the left is a picture of the Néron-Severi group of X in case ρpXq " 3. The green plane is Π f , it intersects the isotropic cone along the two lines Rθ f and Rθ f ; the brown line is its orthogonal complement Π K f , the magenta point is rκ 0 s. If f preserves a curve E, its class is on Π K f . On the right is a projective view of the same picture, but now the two brown lines are the projectivization of the planes pθ f q K and pθ f q K . so in particular, 0 ă xm f |m f y ď 1, and the right hand inequality is an equality if and only if m f " rκ 0 s ( 3 ). In H X , the geodesic joining Ppθ f q and Ppθ f q is the curve Axpf q parametrized by sθ f `tθ f with s P R ˚and st " xθ f |θ f y ´1. The projection of rκ 0 s on Axpf q is ? 2 This can be obtained from elementary Euclidean geometry in the hyperplane x¨|rκ 0 sy " 1 by fixing coordinates in which the quadratic form associated to the intersection product expresses as x 2 0 ´x2 1 .

xθ f |θ f y 1{2 m

. . ´x2

n and rκ 0 s " p1, 0, . . . 0q.

and (ii) the intersection form induces a Minkowski form. Moreover, Π Γ " Π Γ 0 for any finite index subgroup of Γ.

Various sufficient conditions on a subgroup Γ imply that it is non-elementary:

-Γ contains a pair of loxodromic elements pf, gq with θ f , θ f ( X θ g , θ ǵ ( " H; -Γ contains two parabolic elements associated to different fibrations; -Γ contains a parabolic and a loxodromic element.

If AutpXq contains a non-elementary group Γ, then X is automatically projective and Π Γ is contained in the Néron-Severi group NSpX; Rq (see [18, §3.6]). If in addition Γ contains a parabolic element, then Π Γ is defined over Q with respect to the lattice NSpX; Zq (see [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Lem. 2.9]).

The limit set LimpΓq Ă BH X is the closure of the set of fixed points of loxodromic elements in PpΠ Γ q, or equivalently the smallest closed invariant subset in BH X . The following lemma is well-known (see [START_REF] Kapovich | Hyperbolic manifolds and discrete groups[END_REF]Lem. 3.24]):

Lemma 3.3. If Γ is non-elementary, pPpθ f q, Ppθ f qq ; f P Γ lox ( is dense in LimpΓq 2 .
3.2. Periodic curves of loxodromic automorphisms. Our purpose in this paragraph is to bound the degrees of the periodic curves of a loxodromic automorphism.

Lemma 3.4. Let e be an element of H 1,1 pX; Rq such that e is orthogonal to m f and xrκ 0 s|ey " 1. Then xe|ey ă 0 and

´xe|ey ě xm f |m f y 1 ´xm f |m f y " ˆtan ˆ1 2 Ang κ 0 pθ f , θ f q ˙˙2 .
Note that under the assumption of the lemma m f cannot be equal to rκ 0 s, so 0 ă xm f |m f y ă 1 by Remark 3.2.

Proof. Write m f " rκ 0 s `v and e " rκ 0 s `w where v and w are in the orthogonal complement rκ 0 s K . Then, xe|m f y " 0, so xv|wy " ´1, and the Cauchy-Schwarz inequality gives 1 ď p´xv|vyqp´xw|wyq because the intersection form is negative definite on rκ 0 s K . This inequality is equivalent to 1 ď p1 ´xm f |m f yqp1 ´xe|eyq and the result follows.

If C Ă X is a curve, define its degree (with respect to κ 0 ) to be:

degpCq " ż C κ 0 " xrCs|rκ 0 sy, (3.5)
and similarly define the degree of an automorphism g P AutpXq by:

degpgq " ż X κ 0 ^g˚κ 0 " xrκ 0 s|g ˚rκ 0 sy. (3.6)
In the following lemma, K X denotes the canonical bundle of X:

Lemma 3.5. Let c X ě 0 be a constant such that xK X |¨y ď c X xrκ 0 s|¨y on the effective cone. If f P AutpXq is loxodromic and E is a reduced, connected, and f -periodic curve, then xθ f |θ f y degpEq ď 2p1 `cX q. If E is not connected, then E has at most ρpXq ´2 connected components, thus

xθ f |θ f y degpEq ď 2pρpXq ´2qp1 `cX q ď 2pb 2 pXq ´2qp1 `cX q
where ρpXq is the Picard number of X and b 2 pXq is its second Betti number.

If E is f -invariant, then rEs is orthogonal to Π f for the intersection form, so the Hodge index theorem implies that rEs 2 ă 0. Thus, if E is irreducible, it is determined by its class rEs, and Lemma 3.5 shows that f has only finitely many irreducible periodic curves; this finiteness result strengthens [41, Prop. B] (see also [START_REF] Cantat | Dynamique des automorphismes des surfaces K3[END_REF] and [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]Prop. 4.1]). We shall denote by D f the union of these irreducible f -periodic curves.

Example 3.6. We can take c X " 0 when X is a K3, Enriques, or abelian surface.

Proof of Lemma 3.5. Assume first that E is connected. Set e " rEs degpEq so that xe|rκ 0 sy " 1. Since E is reduced and connected, its arithmetic genus xK X `E|Ey `2 is nonnegative (see [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]§II.11]), so

´xE|Ey ď 2 `xK X |Ey ď 2 `cX degpEq. (3.7)
On the other hand Lemma 3.4 implies

´xE|Ey " ´degpEq 2 xe|ey ě xm f |m f y 1 ´xm f |m f y degpEq 2 . (3.8)
Putting these two inequalities together we get

degpEq 2 ď 1 ´xm f |m f y xm f |m f y p2 `cX degpEqq . (3.9)
Solving for the corresponding quadratic equation in degpEq, and applying the inequality tp1 ´tq ď 1{4 with t " xm f |m f y finally gives

xm f |m f y degpEq ď p1 ´xm f |m f yqc X `1{ ? 2 ď c X `1. (3.10)
For the second assertion, write E as a union of disjoint connected components E i . The classes rE i s are pairwise orthogonal, and are contained in pθ f q K Xpθ f q K , a subspace of codimension 2 in the Néron-Severi group of X. This implies that there are at most ρpXq ´2 connected components. Proposition 3.7. Let X be a compact Kähler surface endowed with a reference Kähler form κ 0 such that ş κ 2 0 " 1. If f P AutpXq is loxodromic and E is an f -invariant curve, then degpEq ď 2 54 pρpXq ´2qp1 `cX q degpf q 56 , where the degrees are relative to κ 0 and c X is as in Lemma 3.5.

Proof. As in Remark 3.2, denote by d H the hyperbolic distance on H X and let Axpf q be the axis of the loxodromic isometry f ˚. Lemma 4.8 in [START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF] implies that( 4) d H prκ 0 s, Axpf qq ď 28 d H prκ 0 s, f ˚rκ 0 sq " 28 cosh ´1pdegpf qq.

Then, using the formula (3.4) for the distance d H prκ 0 s, Axpf qq together with the elementary inequality coshpkxq ď 2 k´1 coshpxq k , we obtain

2 xθ f |θ f y " coshpd H prκ 0 s, Axpf qqq 2 ď 2 54 pdegpf qq 56 . (3.11)
The result now follows from Lemma 3.5. it coincides with Π Γ (see §3.1.3). So, rCs P Π K Γ , and in particular rCs 2 ă 0. Let us prove the second assertion. If rCs P Π K Γ and g P Γ par , rCs intersects trivially the class rF s of the general fiber of π g ; this implies that C is contained in a fiber of π g , and is a component of a singular fiber since rCs 2 ă 0. Now, denote by S the set of irreducible curves which are a component of π g for all g P Γ par ; it remains to prove that each C P S is Γ-periodic. Since Γ is non-elementary, Γ par contains two elements g 1 and g 2 with distinct fixed points on the boundary of H X ; these fixed points are respectively given by the classes rF 1 s and rF 2 s of any smooth fiber of π g 1 and π g 2 ; hence, π g 1 and π g 2 can not share any smooth fiber. This shows that elements of S are contained in singular 4 It was stated for birational transformations of P 2 in [START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF] but the estimate holds in our setting with the same proof (actually an easier one since here we work in a finite dimensional hyperbolic space). fibers of π g 1 , and in particular S is finite. Moreover, S is Γ-invariant, because Γ par is invariant under conjugacy, thus every C P S is a Γ-periodic curve.

The following proposition shows that examples as in [19, §11] do not appear for non-elementary groups containing parabolic automorphisms. Proposition 3.9. Let Γ Ă AutpXq be a non-elementary subgroup containing parabolic automorphisms. There is a birational morphism π 0 : X Ñ X 0 onto a normal projective surface X 0 and a homomorphism τ : Γ Ñ AutpX 0 q such that (1) π 0 contracts all Γ-periodic curves and only them;

(2) π 0 is equivariant: π 0 ˝f " τ pf q ˝π0 for every f P Γ;

(3) there is an ample line bundle A on X 0 such that π 0 A is a big and nef line bundle, whose class belongs to Π Γ .

Before starting the proof, recall that a line bundle M on X is semi-ample if and only if M bm is globally generated (or equivalently base-point free) for some m ą 0 (see [45, §2.1.B]). Set FreepX; M q " tm P N | mM is base point freeu. (here we use the additive notation mM for the line bundle M bm .) This defines a semigroup in N, and we denote by frpM q the largest natural number such that every element of FreepX; M q is a multiple of frpM q. Given k in FreepX; M q, the line bundle kM determines a morphism Φ kM : X Ñ X kM Ă PpH 0 pX, kM q _ q, onto a projective (possibly singular) normal variety X kM . According to Theorem 2.1.27 in [START_REF] Lazarsfeld | Positivity in algebraic geometry. I[END_REF], there is an algebraic fibre space Φ : X Ñ Y such that (1) Y is a normal projective variety (see Example 2.1.15 in [START_REF] Lazarsfeld | Positivity in algebraic geometry. I[END_REF]);

(2) X kM " Y and Φ kM " Φ for sufficiently large elements k of FreepX; M q;

(3) there is an ample line bundle A on Y such that Φ ˚A " frpM qM .

Note that conversely the pull-back of a base point free line bundle by a morphism is base point free.

Example 3.10. To each g P AutpXq par corresponds a semi-ample line bundle L g such that (i) the members of |L g | are given by the fibers of π g and (ii) π g : X Ñ B g coincides with the fibration Φ : X Ñ Y determined by L g . The ray R `rL g s Ă H 1,1 pX; Rq determines the unique fixed point of g ˚in BH X , and L g is nef (see Section 3.1).

Proof of Proposition 3.9. By Lemma 3.8 we can fix a finite number of parabolic elements g i P Γ, 1 ď i ď k, such that the set of irreducible and Γ-periodic curves C Ă X is exactly the set of irreducible curves which are contained in fibers of π g i for i " 1, . . . , k. The line bundle M " ř i L g i is semi-ample, it is nef because the L g i are and it is big because M 2 ą 0, finally its class belongs to Π Γ because the classes rL g i s belong to the limit set of Γ (see [18, §2.3.6]). Since M is big, the fibration Φ " Φ kM : X Ñ Y defined, as above, by the sufficiently large multiples of M is a birational morphism (a generically finite fibration is a birational morphism since its fibers are, by definition, connected). By construction Φ contracts exactly the periodic curves of Γ. So, setting π 0 " Φ and X 0 " Y , we obtain a birational morphism that contracts all periodic curves, and only them. Since Γ permutes these curves, it induces a group of automorphisms on X 0 . Moreover, we know that there is an ample line bundle A on X 0 such that π 0 A " frpM qM ; this proves the third assertion.

Remark 3.11. In the proof of Proposition 3.9, one may add extra parabolic automorphisms g j P Γ par , say with k `1 ď j ď , and replace M by ř i"1 m i L i for any choice of integers m i ą 0, while getting the same conclusion. After multiplication by Q ˚, the classes constructed in this way form a dense subset of the convex cone

# ÿ i"1 α i c 1 pL g i q ; ě 1, g i P Γ par , and α i P R ˚for all i + . (3.12)
This cone is Γ-invariant, its closure is the smallest convex cone whose projectivization contains the limit set LimpΓq, and it spans Π Γ because Π Γ is the smallest vector space containing LimpΓq. Thus, the classes of the form αc 1 pπ 0 Aq, where A runs over the set of ample line bundles on X 0 and α runs over Q ˚, is a dense subset of this cone. If AutpXq 0 is non-trivial, then X is a torus and then f has no invariant curve (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]Remark 3.3] and [START_REF] Cantat | Symétries birationnelles des surfaces feuilletées[END_REF]). Otherwise, the kernel of the homomorphism Γ Ñ Γ ˚is finite, f Z has finite index in Γ, and therefore a curve is Γ-periodic if and only if it is f -periodic, so we are done when Γ is elementary. When Γ is non-elementary, Theorem D is covered by the following more precise statement (recall that X is automatically projective in this case [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Thm. 3.17]). Proposition 3.12. Let X be a complex projective surface and Γ be a non-elementary subgroup of AutpXq. Then there exists a loxodromic element f in Γ such that every f -periodic curve is Γ-periodic. If in addition Γ contains a parabolic element, f can be chosen of the form h ˝g, where g and h are parabolic and unipotent.

Proof. Consider a subset S Ă Γ lox such that pPpθ f q, Ppθ f qq ; f P S ( is dense in LimpΓq 2 , as in Lemma 3.3. Let us exhibit an f P S such that every Γ-periodic curve is f -periodic. By contradiction, we assume that every f P S admits at least one irreducible periodic curve Cpf q which is not Γ-periodic, and we set cpf q " rCpf qs. By Lemma 3.8, cpf q does not belong to Π K Γ , thus u Þ Ñ xcpf q|uy is a non-trivial linear form on Π Γ . Since the class of any periodic curve is orthogonal to Π f , xcpf q|θ f y " xcpf q|θ f y " 0.

Let U and U 1 be open subsets of BH X intersecting LimpΓq non trivially, and such that U X U 1 " H; let x be an element of U X LimpΓq. Define ApU, U 1 q " tf P AutpXq ; f is loxodromic, Ppθ f q P U and Ppθ f q P U 1 u. and DpU, U 1 q " tcpf q ; f P ApU, U 1 q X Su.

(3.13) By Lemma 3.5, DpU, U 1 q is a finite set. From our assumption on S, there is a sequence pf n q of elements in ApU, U 1 q X S such that x " lim n pPpθ fn qq. Extracting a subsequence if necessary we may assume that cpf n q is constant, equal to some c 8 P DpU, U 1 q, and we infer that x is contained in c K 8 . As a consequence, the limit set LimpΓq Ă PpΠ Γ q is locally contained in the finite union of hyperplanes Ppc K X Π Γ q, for c P DpU, U 1 q. By compactness, LimpΓq is contained in a finite union of hyperplanes, which contradicts the strong irreducibility of the action of Γ on Π Γ . Now, to prove the first assertion of the proposition, we simply put S " Γ lox , which satisfies the desired density property by Lemma 3.3. For the second assertion, we let S be the set of loxodromic elements of the form h ˝g, where g and h are parabolic and unipotent. To check the density property, we first observe that the set of fixed points of parabolic elements is dense in LimpΓq: indeed it is enough to consider the conjugates of a single parabolic transformation. Then, applying the next lemma together with Lemma 3.1 finishes the proof. Lemma 3.13. Let h and h 1 be two parabolic elements of AutpXq with distinct fixed points u and u 1 in BH X . Let U and U 1 be small, disjoint neighborhoods of u and u 1 , respectively, in PpΠ Γ q. Then if N P Z is large enough, f N :" h N ˝ph 1 q N is a loxodromic automorphism such that Ppθ fN q P U and Ppθ fN q P U 1 .

Proof. Let us denote by Pph ˚q the linear projective transformation induced by h ˚on PpNSpX; Rqq. Since U does not contain u 1 , Pph 1˚qN pU q Ă U 1 if |N | is large enough; similarly Pph ˚qN pU 1 q Ă U . So for f N " h N ˝ph 1 q N , Ppf N q maps U 1 stricly inside itself and likewise Pppf ´1 N q ˚q maps U strictly inside itself. This implies that f N is loxodromic, with its α-limit and ω-limit points in U 1 and U respectively.

COMPLEX TORI AND KUMMER EXAMPLES

This section gathers some facts on automorphism groups of complex tori. We also introduce and study the notion of Kummer group. Part of this material is well-known, we provide the details for completeness. 4.1. Finite orbits on tori. Consider a compact complex torus A " C k {Λ. Each automorphism f of A is an affine transformation f pzq " L f pzq `tf , where z Þ Ñ z `tf is the translation part and L f is a linear automorphism, induced by a linear transformation of C k that preserves Λ. Let Γ be a subgroup of AutpAq.

Warning. By definition, compact tori and abelian varieties come equipped with their group structure, in particular with their neutral element, or "origin". On the other hand, an automorphism f with a non-trivial translation part t f does not preserve this group structure. If x P A is fixed by Γ, conjugating Γ by the translation z Þ Ñ z `x we may assume that Γ fixes the neutral element of A and acts by linear isomorphisms. Alternatively, we can transport the group structure by using this translation and put the neutral element at x: this changes the group structure without changing the underlying complex manifold. We will frequently do this operation in the following, without always specifying the change in the group structure of A.

Suppose that the orbit Γpxq Ă A is finite, of cardinality m, and consider the stabilizer Γ 0 Ă Γ of x; its index divides m!. Conjugating Γ by z Þ Ñ z `x, as explained above, all elements f P Γ 0 are linear. In that case, every torsion point has a finite Γ 0 -orbit, hence also a finite Γ-orbit; in particular, finite orbits of Γ form a dense subset of A for the Euclidean topology. The next proposition summarizes this discussion. Proposition 4.1. Let A be a compact complex torus, and let Γ be a subgroup of AutpAq. If Γ has a finite orbit, then its finite orbits form a dense subset of A. More precisely if a periodic point of Γ is chosen as the origin of A for its group law, then all torsion points of A are periodic points of Γ. Remark 4.2. If in Proposition 4.1 we moreover assume that dim C A " 2 and Γ contains a loxodromic element, then conversely all periodic points of Γ are torsion points. This follows from Lemma 4.3 below. [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF][START_REF] Curtis | Dynamics on K3 surfaces: Salem numbers and Siegel disks[END_REF]). Let A " C 2 {Λ be a compact complex torus of dimension 2, and let f pzq " L f pzq `tf be a loxodromic element of AutpAq. The loxodromy means exactly that the eigenvalues α and β of L f satisfy |α| ă 1 ă |β|. Pick a basis of Λ, and use it to identify Λ with Z 4 and C 2 with R 4 , as real vector spaces. Then, L f corresponds to an element M f P GL 4 pZq. Lemma 4.3. Let f be a loxodromic automorphism of a compact complex torus A of dimension 2. Then:

Dimension 2 (see

(1) f has a fixed point, and after translation z Þ Ñ z `x by such a fixed point, its periodic points are exactly the torsion points of A; (2) there is no f -invariant curve: the orbit f Z pCq of any curve is dense in A.

Proof. For (1), using the above notation, fixed points of f are determined by the equation pL f ´idqpzq P Λ ´tf , or equivalently pM f ´idqpzq P Z 4 ´tf . Since the complex eigenvalues of L f are distinct from 1, there is at least one fixed point. So, after conjugation by a translation, we may assume that t f " 0. Then, periodic points of f correspond to points x P R 4 such that M n f pxq ´x P Z 4 : any solution to such an equation is rational, which means that it corresponds to a torsion point in A.

To prove (2) without loss of generality we may assume that t f " 0. There are two linear forms ξ f , ξ f : C 2 Ñ C such that ξ f ˝Lf " αξ f and ξ f ˝Lf " βξ f . They determine two holomorphic 1-forms on A, the kernels of which define two linear foliations: the stable and unstable foliations of f . Both have dense leaves. The class θ f is represented by ξ f ^ξf , up to some positive multiplicative factor. If C Ă A is a complex analytic curve, there is an open subset U of A in which C intersects both foliations transversely. If y is a torsion point, y is f -periodic, and its stable manifold being dense, it intersects C. Thus, f n pCq accumulates every torsion point, and is dense in A. In particular, C is not invariant, as claimed.

Kummer structures.

4.3.1. Kummer pairs. Let X be a compact complex surface, and let Γ be a subgroup of AutpXq. By definition pX, Γq is a Kummer group if there is an abelian surface A, a finite subgroup G of AutpAq, a subgroup Γ A of AutpAq containing G, and a birational morphism q X : X Ñ A{G such that: (a) Γ A normalizes G. Thus, if q A : A Ñ A{G is the quotient map, there is a homomorphism h P Γ A Þ Ñ h P AutpA{Gq such that q A ˝h " h ˝qA for every h P Γ A ; we shall denote by Γ A the image of this homomorphism. (b) the birational map q X is Γ-equivariant: there is a homomorphism

Γ Q f Þ Ñ f P
AutpA{Gq, whose image is denoted by Γ, such that q X ˝f " f ˝qX for every f P Γ. (c) the subgroups Γ and Γ A of AutpA{Gq coincide.

To each f P Γ corresponds an element f A of Γ A , unique up to composition with elements of G; the type of f as an automorphism of AutpXq coincides with the type of f A as an automorphism of A, and λpf q " λpf A q. Remark 4.4. Consider a section Ω A of the canonical bundle K A such that ş A Ω A ^ΩA " 1; it is unique up to multiplication by a complex number of modulus one. In particular, the volume form vol A " Ω A ^ΩA is invariant under AutpAq. The quotient of vol A by the action of G determines a probability measure on A{G, and then on X. This probability measure coincides with the measure of maximal entropy µ f for every f P Γ lox .

From the definition of a Kummer group, Proposition 4.1 and Remark 4.2 we get: Proposition 4.5. If pX, Γq is a Kummer group with at least one finite orbit, then its finite orbits are dense in X for the euclidean topology. Furthermore there exists a dense, Γinvariant, Zariski open subset in which all periodic points of loxodromic elements of Γ coincide.

Classification of Kummer examples.

Let us consider, firstly, the case of an infinite cyclic group generated by a loxodromic Kummer example f . From the classification given in [START_REF] Cantat | Symétries birationnelles des surfaces feuilletées[END_REF][START_REF] Cantat | Corrigendum: "Birational symmetries of foliated surfaces[END_REF], we may assume that the finite group G is a cyclic group fixing the origin of A; in other words, G is induced by a cyclic subgroup of GL 2 pCq preserving the lattice Λ such that A " C 2 {Λ. And there are only seven possibilities:

(1) G " tidu and X is a blow-up of an abelian surface.

(2) G " tid, ´idu and A{G is a Kummer surface, in the classical sense; in particular X is a blow-up of a K3 surface. (3) A is the torus pC{Zrisq 2 and G is the group of order 4 generated by iid ; in this case X is a rational surface. (4) A is the torus pC{Zrexpp2iπ{3qsq 2 and G is the group of order 3 generated by expp2iπ{3qid; in this case X is a rational surface. (5) A is the torus pC{Zrexpp2iπ{3qsq 2 and G is the group of order 6 generated by exppiπ{3qid; in this case X is a rational surface. (6) Let ζ 5 be a primitive fifth root of unity. The cyclotomic field Qrζ 5 s has two distinct non-conjugate embeddings in C, σ 1 and σ 2 determined by σ 1 pζ 5 q " ζ 5 and σ 2 pζ 5 q " ζ 2 5 . The ring of integers coincides with Zrζ 5 s and its image by σ " pσ 1 , σ 2 q is a lattice Λ 5 Ă C ' C. The abelian surface A is the quotient C 2 {Λ 5 . The group G is generated by the diagonal linear map px, yq Þ Ñ pζ 5 x, ζ 2 5 yq (4.1)

and has order 5. Here, X is rational too. (7) As in the previous example, A " C 2 {Λ 5 , but now G has order 10 and is generated by px, yq Þ Ñ p´ζ 5 x, ζ 5 yq, and again X is rational.

These constraints on pA, Gq apply to non-elementary Kummer groups; in particular we shall always assume that G is cyclic and fixes the neutral element of A.

In Cases (1) to (5) of the above list, the abelian surface is C 2 {pΛ 0 ˆΛ0 q for some lattice Λ 0 in C. The natural action of GLp2, Zq on C 2 preserves Λ 0 ˆΛ0 , and induces a non-elementary subgroup of AutpAq, which commutes to G; as a consequence, it determines also a non-elementary subgroup of AutpA{Gq. On the other hand, cases ( 6) and ( 7) do not appear: Lemma 4.6. If pX, Γq is a non-elementary Kummer group, then G is generated by a homothety and the quotient A{G is not of type ( 6) or [START_REF] Birkenhake | Complex abelian varieties, volume 302 of Grundlehren der Mathematischen Wissenschaften[END_REF] in the classification above.

Proof. The group Γ A permutes the fixed points of G. So, the stabilizer Γ A " Stab Γ A p0q of the neutral element is a finite index, non-elementary subgroup of Γ A . Pick any loxodromic element f in Γ A; it acts by conjugacy on G, which is finite, so there is a positive iterate such that f n ˝g " g ˝f n for all g P G. Near the orgin, f n and g are two commuting linear transformations, f n has two eigenvalues, of modulus ă 1 and ą 1 respectively, and g must preserve the corresponding stable and unstable directions of f . Since Γ A is non-elementary, these tangent directions form an infinite set as f varies in the set of loxodromic elements of Γ A, so g is a homothety, and we are done. 4.3.3. Invariant curves. We keep the notation from the previous paragraphs and consider a non-elementary Kummer group pX, Γq. The singularities of A{G are cyclic quotient singularities, and X dominates a minimal resolution of A{G.

Let us examine Case (2), when G " tid, ´idu. Then G has 16 fixed points, and to resolve the 16 singularities of A{G one can proceed as follows. First, one blows up the fixed points, creating 16 rational curves. Then one lifts the action of G to the blow-up Â. If E is one of the exceptional divisors, then G fixes E pointwise and acts as w Þ Ñ ´w transversally, so locally the quotient map can be written pw, zq Þ Ñ pw 2 , zq, with E " tw " 0u giving rise to a smooth rational curve of self-intersection ´2 on Â{G. This construction provides the minimal resolution X min " Â{G of A{G, the singularities being replaced by disjoint p´2q-curves. Cases (3), ( 4), ( 5) can be handled with a similar process because if x P A is stabilized by a subgroup H of G, then H is locally given around x as a cyclic group of homotheties; so, in the minimal resolution of A{G the singularities are replaced by disjoint rational curves E i of negative self-intersection E 2 i P t´2, . . . , ´6u. Cases ( 6) and ( 7) are more delicate however, by Lemma 4.6, we don't need to deal with them (see Appendix A for Case (6)).

Lemma 4.7. Let pX, Γq be a non-elementary Kummer group on a smooth projective surface. Then:

(1) X is abelian if and only if Γ admits no invariant curve;

(2) any connected Γ-periodic curve D is a smooth rational curve, and the induced dynamics of Stab D pΓq on D has no periodic orbit.

Moreover, D f " D Γ for every f P Γ lox .

Proof. The minimal resolution X min of A{G is unique, up to isomorphism (see [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]§III.6], Theorems (6.1) and (6.2), and their proofs). Thus, X dominates X min and every f P Γ preserves the exceptional divisor of the morphism X Ñ X min and induces an automorphism f min of X min . In particular Γ admits an invariant curve, unless G " tidu and X " X min " A. Conversely in that case Γ has no invariant curve, by Lemma 4.3. This proves the first assertion.

Let us prove the second assertion for the induced group Γ min Ă AutpX min q. Let E be a connected periodic curve for Γ min . If f min P Γ min is loxodromic, it comes from an Anosov map f A : A Ñ A, as in Lemma 4.3, and f A does not have any periodic curve. Since E is f min -periodic, it is contained in the exceptional divisor of the resolution X min Ñ A{G; as explained before the lemma, this divisor is a disjoint union of rational curves, so E is one of these rational curves E x " q ´1 X min pq A pxqq, where x P A has a nontrivial stabilizer G x Ă G. In particular x is fixed by a finite index subgroup Γ A,x of Γ A . Now since Γ is non-elementary, Γ A and Γ A,x are non-elementary as well, and since the action of Γ A,x on A is by affine transformations, its action on the exceptional divisor E x is that of a non-elementary subgroup of PGL 2 pCq " AutpE x q. In particular, Γ A,x does not admit any finite orbit in E x .

The birational morphism π : X Ñ X min is equivariant with respect to Γ and its image Γ min in AutpX min q. So, π ´1 blows up periodic orbits of Γ min . The last few lines show that, when such a periodic point y P X min is blown up, firstly y does not lie on the exceptional locus of X min Ñ A{G, and secondly the exceptional divisor E y does not contain any finite orbit. So, X is obtained by simple blowups centered on a finite set of distinct periodic points of Γ min , every connected component of the exceptional locus of q X is a smooth rational curve, and there is no Γ-periodic point in these curves.

UNLIKELY INTERSECTIONS FOR NON-ELEMENTARY GROUPS

This section is devoted to the proof of Theorem B. Since the proof comprises many steps, let us start with a rough outline of the argument. 5.1. Strategy of the proof. If Γ has a Zariski dense set F of finite orbits, a standard argument shows that there exists a sequence px n q P F N which is generic in the sense that it escapes any fixed proper subvariety Y Ă X. Let m xn be the probability measure equidistributed over the Galois orbit of x n . We want to use arithmetic equidistribution to show that the sequence of measures pm xn q converges when n Ñ 8. For this, we need a height function h L associated to some line bundle L on X satisfying appropriate positivity properties, and such that h L vanishes on F . In § §5.2-5.4 we construct such height functions: they are associated to the choice of certain finitely supported probability measures ν on Γ. Indeed to such a measure, we associate the linear endomorphism P ν " ř νpf qf ˚on the Néron-Severi group of X, and we construct a big and nef line bundle L such that P ν rLs " αpνqrLs for some αpνq ą 1; then, h L will be a Weil height that satisfies the invariance ř νpf qh L ˝f " αpνqh L . The arithmetic equidistribution theorem of Yuan shows that the measures m xn converge to a measure µ ν " S ν ^Sν , where S ν is a dynamically defined closed positive current with cohomology class equal to rLs. On the other hand, the measures m xn , hence their limit µ, do not depend on ν. As we vary the choice of ν, there is enough freedom in the construction to show that for every f P Γ lox , µ ν can be made arbitrary close to the maximal entropy measure µ f . It follows that µ f " µ is independent of f and is Γ-invariant. In §5.5, using the dynamics of parabolic elements of Γ we deduce that Supppµq " X. Then the classification of Γ-invariant measures from [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Cantat | Dynamics of automorphism groups of complex projective surfaces: parabolic maps and invariant measures[END_REF] implies that µ has a smooth density, and the main result of [START_REF] Cantat | Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy[END_REF] shows that every f P Γ lox is a Kummer example (Theorem 5.14 in §5.6). At this point the Kummer structure may a priori depend on f , as in Example 5.1 below. This issue is solved in §5.7 by adding an argument based on Theorem D, which finally shows that pX, Γq is a Kummer group.

Example 5.1. Let X be a Kummer surface possessing both a Kummer automorphism f and a non-Kummer one h, as in [START_REF] Keum | The automorphism groups of Kummer surfaces associated with the product of two elliptic curves[END_REF]. Then, f and h ˝f ˝h´1 are two Kummer automorphisms which are not associated with the same Kummer structure; the pair pX, xf, h ˝f ˝h´1 yq is not a Kummer group. 5.2. Kawaguchi's currents.

5.2.1.

Action on H 1,1 . Let X be a Kähler surface and let ν be a probability measure on AutpXq satisfying the (exponential) moment assumption

ż `}f } C 1 `› › f ´1› › C 1 ˘2 dνpf q ă `8. (5.1)
By [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Lem. 5.1] this implies the following moment condition on the cohomological action of Γ:

ż }f ˚}dνpf q ă `8, (5.2)
where f ˚is the endomorphism of H 2 pX; Rq determined by f and }¨} is any operator norm. (For the proof of Theorem 5.14, we will only consider finitely supported measures so the moment conditions will be trivially satisfied.) Let Γ ν be the subgroup of AutpXq generated by the support of ν. We define a linear endomorphism P ν of H 2 pX; Cq by P ν puq " ż f ˚puq dνpf q.

(5.

3)

The following lemma is a strong version of the Perron-Frobenius theorem (see [START_REF] Birkhoff | Linear transformations with invariant cones[END_REF]).

Lemma 5.2. Assume that Γ ν does not fix any isotropic line in H 1,1 pX; Rq. Then, P ν has a unique eigenvector w ν P H 1,1 pX; Rq such that w 2 ν " 1 and xw ν |rκ 0 sy ą 0. This eigenvector is big and nef. The eigenvalue αpνq such that P ν pw ν q " αpνqw ν is larger than 1 and coincides with the spectral radius of P ν ; the multiplicity of αpνq is equal to 1, and all other eigenvalues β P C of P ν satisfy |β| ă αpνq.

Proof. Let u be an isotropic vector, contained in the closure of the positive cone PospXq. Then, P ν puq has positive self-intersection, except if f ˚puq P Ru for ν-almost every f , and then for all f in the support of ν by continuity. Hence, the hypothesis implies that P ν maps the positive cone strictly inside itself. From the Perron-Frobenius theorem ( [START_REF] Birkhoff | Linear transformations with invariant cones[END_REF]), P ν has an eigenvector w ν in the interior of this cone such that (1) w ν is a dominant eigenvector: the eigenvalue αpνq such that P ν pw ν q " αw ν is the spectral radius of P ν .

Since w ν is in the interior of PospXq, we may assume that w ν is in H X . Projectively, PpP ν q contracts strictly the Hilbert metric of the convex set PpPospXqq " PpH X q; so, P ν does not have any eigenvector in H X besides w ν itself, and the PpP ν q-orbit of any point of PpH X q converges towards Ppw ν q. The Kähler cone is also invariant: If w 1 is another eigenvector with eigenvalue αpνq and w 1 R Rw ν , the plane Vectpw 1 , w ν q intersects H X along a geodesic, all of whose elements satisfy P ν pvq " αpνqv. From assertion (2), we get a contradiction. Now, if the multiplicity of αpνq in the characteristic polynomial of P ν is larger than 1, there exists a vector u R Rw ν such that P ν puq " αpνqu `αpνqw ν . In the plane Vectpw ν , uq the cone PospXq is bounded by two rays; changing u into u `cw ν if necessary (for some c P R), these two rays are R `pw ν `auq and R `pw ν ´buq for some positive real numbers a and b. But then, the image of R `pw ν ´buq by P ν is not contained in PospXq, a contradiction. It follows that

P ν pKahpXqq Ă KahpXq. Hence, w ν is nef (it is
(3) αpνq is a simple root of the characteristic polynomial of P ν .

Finally, (4) if P ν pvq " βv is any complex eigenvector with v R Cw ν , then |β| ă αpνq.

Indeed, if β is another eigenvalue with |β| " αpνq, then β " αpνqe 2iπθ for some θ P R, and there is a plane V Ă H 1,1 pX; Rq on which P ν acts as a similitude of strength αpνq and angle 2πθ; pick v in this plane, then if ε P R ˚is small, the vector w ν `εv is in PospXq and the PpP ν q-orbit of Ppw ν `εvq in PpH X q stays at constant distance from Ppw ν q, contradicting the contraction property of P ν . This concludes the proof.

Example 5.3 (See [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF], §2, and [START_REF] Cantat | Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy[END_REF], §2.2). Let f be a loxodromic automorphism, and take ν to be the probability measure pδ f `qδ f ´1 with p, q ě 0 and p `q " 1. Note that Γ ν " f Z does not satisfy the assumption of Lemma 5.2. Then P ν " pf ˚`qpf ´1q ˚preserves the f ˚-invariant plane Π f Ă H 1,1 pX; Rq. If p ą q, the spectral radius of P ν is equal to pλpf q `q{λpf q, and the corresponding eigenspace is the isotropic line of Π f corresponding to the eigenvalue λpf q of f ; if p ă q, the spectral radius is equal to p{λpf q `qλpf q and the eigenspace is the other isotropic line in Π f . If p " q " 1{2, then pP ν q |Π f is the scalar multiplication by αpf q " 1 2 pλpf q `1{λpf qq (5.4) and all vectors u P Π f satisfy P ν puq " αpf qu. This example shows that the previous lemma fails for ν, whatever the values of p and q are: the dominant eigenvector is at the boundary of the hyperbolic space, or is not unique.

Stationary currents.

Let us borrow some notation from [18, §6.1]: we fix Kähler forms κ i , the cohomology classes of which provide a basis prκ i sq of H 1,1 pX; Rq. Then, if a is any element of H 1,1 pX; Rq, there is a unique p1, 1q-form Θpaq " ř i a i κ i in Vectpκ i , 1 ď i ď h 1,1 pXqq whose class rΘpaqs is equal to a. If S is any closed positive current of bidegree p1, 1q, then S " ΘprSsq `dd c u S for some upper semi-continuous function u S : X Ñ R: this function is locally the difference of a plurisubharmonic function and a smooth function, and it is unique up to an additive constant.

The following proposition is essentially due to Kawaguchi, who proved it in [START_REF] Kawaguchi | Canonical heights, invariant currents, and dynamical eigensystems of morphisms for line bundles[END_REF] under slightly more restrictive assumptions.

Proposition 5.4. Let X be a compact Kähler surface, and let vol be a smooth volume form on X. Let ν be a probability measure on AutpXq satisfying the moment condition (5.1). Assume that there exists w P H 1,1 pX; Rq and α ą 1 satisfying (i) P ν pwq " αw;

(ii) w is big and nef and w 2 " 1.

Then, there is a unique closed positive current S ν such that ż f ˚pS ν q dνpf q " αS ν and rS ν s " w.

(5.5)

This current has continuous potentials: S ν " Θpwq `dd c puq for a unique continuous function u such that ş X u vol " 0. In particular, the product S ν ^Sν is a well-defined probability measure on X.

Note that here Γ ν is not assumed to be non-elementary. Actually this proposition will be applied in two situations:

-when Γ ν does not fix any boundary point of H X , w " w ν , and α " αpνq, as in Lemma 5.2; -when ν " 1 2 pδ f `δf ´1 q for some loxodromic automorphism and w is a fixed point of 1 αpf q P ν in H X , as in Example 5.3.

Proof of Proposition 5.4. Fix a Kähler form κ 0 on X, as in §3.1. Let β be a smooth form with rβs " w. For simplicity, we denote by the same letter P ν the operator ş f ˚p¨qdνpf q acting on the cohomology, on differential forms, or on currents. Write pα ´1P ν qβ " β `dd c phq for some smooth function h. Then,

ˆ1 α P ν ˙n β " β `dd c ˜n´1 ÿ j"0 1 α j pP ν q j phq ¸, (5.6)
where pP ν q j phqpxq " ş h˝f pxqdν ‹j pf q is the average of h with respect to the probability measure ş δ f pxq dν ‹j pf q and ν ‹j denotes the j-th convolution of ν. The supremum of |pP ν q j phq| on X satisfies }pP ν q j phq} 8 ď }h} 8 for all j ě 1. From this we deduce that the series on the right hand side of Equation (5.6) converges geometrically:

ÿ j"k › › › › 1 α j pP ν q j phq › › › › 8 ď α ´αk´ α ´1 }h} 8 α k ď α α ´1 }h} 8 α k . (5.7)
Thus, if we set h ν " ř jě0 1 α j pP ν q j phq and S ν " β `dd c ph ν q we see that S ν is a closed current which satisfies P ν pS ν q " αS ν . Furthermore, since β can be written as a difference of positive closed currents (e.g. by writing β " pβ `Cκ 0 q ´Cκ 0 for a suitably large positive C), we infer that S ν " lim nÑ8 pα ´1P ν q n β is also a difference of positive closed currents. Let us prove that S ν is positive and unique.

By (ii), w is nef, which implies that the set Curpw ν q of closed positive currents with class w ν is not empty. The linear operator α ´1P ν preserves this compact convex set; as a consequence, Curpw ν q contains a fixed point T of α ´1P ν . So we can fix a non-zero closed positive current T satisfying P ν pT q " αT and rT s " rS ν s. Since T ´Sν is cohomologous to zero and expresses as a difference T 1 ´T2 of positive closed currents, according to [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Lemma 6.1], we can write T ´Sν " dd c phq where h " u 1 ´u2 , and u i is an pAκ 0 q-psh function (A depends only of the mass of the T i ). Changing u i into u i ´şX u i vol we may assume that ş X u i vol " 0 for i " 1, 2. From the invariance of T ´Sν under α ´1P ν we obtain that

1 α P ν phq " h `c (5.8)
for some constant c P R; thus, α ´nP n ν phq " h `cn where c n converges geometrically towards some real number c 8 . From [18, Lemma 6.6], there is a constant

C ą 1 such that ż X 1 α n |P n ν phq|vol ď C α n ż X logpC › › Jacpf ´1q
› › 8 q dν ‹n pf q (5.9) for all n ě 1. Thanks to the moment condition (5.2) and the subadditivity property

logp › › Jacppf ˝gq ´1q › › 8 q ď logp › › Jacpf ´1q › › 8 q `logp › › Jacpg ´1q › › 8 q,
we see that

ż X logp › › Jacpf ´1q › › 8 q dν ‹n pf q " ż X logp › › Jacpf ´1 n ˝¨¨¨˝f ´1 1 q › › 8 q dνpf 1 q ¨¨¨dνpf n q ď n ÿ j"1 ż X logp › › Jacpf ´1 j q › › 8 q dνpf 1 q ¨¨¨dνpf n q
" Opnq, so the right-hand side of the inequality (5.9) tends to 0 as n goes to `8. Hence α ´nP n ν phq converges towards 0 in L 1 pX; volq. Since α ´nP n ν phq " h`c n also converges towards h `c8 , we deduce that h is a constant, namely h " ´c8 , and that T " S ν . In particular S ν is positive, and is the unique positive closed current satisfying (5.5).

Finally, since S ν has continuous potentials, S ν ^Sν is a well-defined positive measure; its total mass is 1 because 1 " w 2 " rS ν s 2 " ş X S ν ^Sν .

Remark 5.5. Here is another setting, closer to [START_REF] Kawaguchi | Canonical heights, invariant currents, and dynamical eigensystems of morphisms for line bundles[END_REF], in which the same result holds. Let X be a (singular) complex projective surface and L be an ample line bundle on X. Pick any integer m ě 1 such that mL is very ample, and consider the Kodaira-Iitaka embedding Φ mL : X Ñ P N , with P N :" PpH 0 pX, mLq _ q. Then, L coincides with Φ mL pOp1q |X q and any Fubini-Study form ω determines a smooth p1, 1q-form κ mL :" Φ mL pωq on X (see [28, §1] for forms, currents, and potentials on singular complex spaces). The form κ L :" 1 m κ mL is locally equal to dd c pvq for some smooth function, namely v " 1 m u ˝ΦmL where u is a local potential of ω in P N . Now consider a probability measure ν on AutpXq such that P ν L " αL, with α ą 1. Then, the proof of Proposition 5.4 applies, and provides a closed positive current S ν on X with continuous potentials such that P ν S ν " αS ν ; this is proven in [40, Thm. 3.2.1] (Kawaguchi assumes X to be smooth, but this is only used in the first line of the proof to introduce the smooth form κ L ).

Example 5.6. As in Example 5.3, consider the case ν " 1 2 pδ f `δf ´1 q, where f is a loxodromic automorphism. There are two closed positive currents T f and T f , with continuous potentials, such that f ˚pT f q " λpf q ˘1T f ; they are unique up to a positive scalar factor and their classes generate the isotropic lines Rθ f (see [15, §5]). By convention, we choose them so that xrT f s|rT f sy " 1, or equivalently, T `^T ´is a probability measure; to determine them uniquely we further require xrT f s|rκ 0 sy " xrT f s|rκ 0 sy. Beware that this normalization is different from that of θ f so a priori rT f s ‰ θ f . Pick a class w " arT f s `brT f s with a, b ą 0 such that w 2 " 1 (equivalently 2ab " 1). By uniqueness, the current S ν provided by Proposition 5.4 is equal to aT f `bT f ; in particular, the measure S ν ^Sν is equal to T f ^T f which, in turn, is the measure of maximal entropy µ f (see §3.1 and [15, § §5.2, 8.2]). 5.2.3. Continuity properties of stationary currents. Now, consider a sequence of probability measures pν n q such that the support of each ν n is contained in a fixed finite set tf 1 , . . . , f m u:

ν n " ÿ i ν n pf i qδ f i , (5.10)
with coefficients in the simplex of dimension m ´1 determined by the constraints ν n pf i q ě 0 and ř i ν n pf i q " 1. Assume that pν n q converges towards ν 8 " ř i ν 8 pf i qδ f i , and that none of the groups Γ νn fixes a point in BH X . For n P N, we denote by w νn P H X the eigenvector of the operator P νn given by Lemma 5.2, and by S νn the current given by Proposition 5.4 for the class w νn ; we shall write S νn " Θpw νn q `dd c u νn , as in Proposition 5.4. For the measure ν 8 , we make one of the following two assumptions:

(a) Γ ν8 does not fix any point of BH X Ă PpH 1,1 pX; Rqq; or (b) ν 8 " 1 2 pδ f `δf ´1 q for some loxodromic automorphism f , and w νn converges to w ν8 :" 1 ? 2 prT f s `rT f sq, with notation as in Example 5.6. In both cases, Proposition 5.4 provides a unique closed positive current S ν8 with class rS ν8 s " w ν8 such that P ν8 " αpν 8 qS ν8 ; it coincides with 1 ? 2 pT f `T f q in case (b). In case (a), by the uniqueness in Lemma 5.2, the classes w νn converge towards w ν8 ; in case (b) this convergence holds by assumption. Note that the corresponding constants αpν n q converge as well.

Lemma 5.7. Under the above assumptions, (1) the sequence of closed positive currents pS νn q converges towards S ν8 ;

(2) the canonical (continuous) potentials u νn converge uniformly to that of S ν8 ;

(3) the sequence of measures µ n :" S νn ^Sνn converges towards S ν8 ^Sν8 .

In case (b), S ν8 ^Sν8 is the unique measure of maximal entropy µ f of f . Proof. The first assertion follows from the uniqueness of the current S ν , obtained in Proposition 5.4, and the compactness of the space of currents of mass 1. The geometric convergence obtained from Equation (5.7) shows that the sequence of potentials u νn is equicontinuous, and by the uniqueness of the normalized potentials, it follows that pu νn q converges uniformly to u ν8 . Then the convergence of the sequence of measures pµ n q follows from the continuity properties of wedge products of currents (see [START_REF] Demailly | Complex analytic and differential geometry[END_REF]III.3.6]). Finally, the characterization of S ν8 ^Sν8 in case (b) was explained in Example 5.6.

Rational invariant classes.

We now construct sequences of probability measures for which the fixed classes w νn have good positivity and integrality properties; the last assertion makes use of the contraction π 0 : X Ñ X 0 constructed in Proposition 3.9.

Proposition 5.8. Let Γ be a non-elementary subgroup of AutpXq such that Π Γ is defined over Q. Let f be a loxodromic element of Γ. There is a sequence pν n q of probability measures on AutpXq such that (1) The support Supppν n q is a finite subset F of Γ that does not depend on n and generates a non-elementary subgroup of Γ containing f ; (2) ν n pgq is a positive rational number for all g P F ;

(3) the unique eigenvector w νn of P νn in H X is an element of R `NSpX ; Zq; (4) the corresponding eigenvalue αpν n q belongs to Q `Xs1, `8r;

(5) ν n converges to the measure 1 2 pδ f `δf ´1 q and w νn converges to 1 ? 2 prT f s `rT f sq. If Γ contains a parabolic element g, one can furthermore assume that g belongs to F and that w νn P R `π0 rA n s for some ample line bundle A n on X 0 .

Proof. For the proof we use the conventions of §3.1.2, in particular the classes θ f , which can be defined by θ f " xT f |rκ 0 sy ´1rT f s.

Step 1.-Since the representation of Γ on Π Γ is irreducible, it is also irreducible over C. Indeed, if W is a proper, Γ-invariant, complex subspace of Π Γ b R C, then W does not contain any non-zero real vector u P H 1,1 pX; Rq; in particular, it does not contain any isotropic eigenvector of any loxodromic element of Γ. This implies that W is contained in the orthogonal complement pθ h q K for all h P Γ lox . But in Π Γ the intersection Ş hPΓ lox pθ h q K is defined over R and is Γ-invariant, so it is trivial. Thus, according to Burnside's theorem (see [START_REF] Halperin | Burnside's theorem on algebras of matrices[END_REF]), Γ contains a basis of the real vector space EndpΠ Γ q. More precisely, one can find a basis pf 1 , f 2 , . . . , f N q with N " pdim Π Γ q 2 such that f i P Γ for all i, f 1 " f and f 2 " f ´1 (indeed, f and f ´1 are linearly independent endomorphisms). In particular, the set of linear combinations ř i α i f i with α i ě 0 contains a non-empty, open, and convex cone of EndpΠ Γ q.

If Γ contains a parabolic element g we can further require that g belongs to the basis, because f ˚, g ˚, and pf ´1q ˚are linearly independent, as can be seen by diagonalizing f ˚.

Step 2.-Set F " tf 1 , f 2 , ¨¨¨, f N u and ∆ N " tpν i q P R N `; ř i ν i " 1u. Let ∆ N be the interior of this simplex. Points in ∆ N correspond to probability measures ν " ř i ν i δ f i whose support is equal to F . When ν P ∆ N , the group Γ ν is non-elementary by Step 1, so by Lemma 5.2 P ν has a unique fixed point w ν in H X . As a consequence, the map ν P ∆ ˝Þ Ñ w ν is continuous. Now, consider a sequence pν n q of elements of ∆ N converging to aδ f `p1 ´aqδ f ´1 , with 0 ă a ă 1. Normalize the fixed point w νn by setting w n :" xw νn |rκ 0 sy ´1w νn , so that xw n |rκ 0 sy " 1 and w n stays in a compact subset of H 1,1 pX; Rq. If w n j converges to w along a subsequence pn j q the limit is a nef eigenvector of the operator af ˚`p1 áqpf ´1q ˚associated to an eigenvalue ě 1. Thus, if a is small the limit must be equal to θ f and the sequence pw n q converges towards this eigenvector (see Example 5.3). Conversely, if 1 ´a is small, then the limit is θ f . The subset ∆ N pεq " tpν i q P ∆ N ; ν i ď ε, @i ě 3u (5.11) is connected. So, the closure of its image by the continuous map ν Þ Ñ xw ν |rκ 0 sy ´1w ν is a compact and connected subset of Π Γ , and the intersection of these compact sets is also connected. This set is contained in the segment rθ f , θ f s because it is contained in PospXq and in the union of eigenvectors of af ˚`p1 ´aqpf ´1q ˚, for a P r0, 1s. Since it contains the endpoints of this segment, it actually coincides with it.

From this we deduce that there exists a sequence of probability measures ν n P ∆ N such that xw νn |rκ 0 sy ´1w νn converges to the class 1 2 pθ f `θf q, hence w νn converges to the class 1 ? 2 prT f s `rT f sq. Then the arguments given in Example 5.3, Example 5.6, and Lemma 5.7 show that:

-P νn converges towards 1 2 pf ˚`pf ´1q ˚q; -αpν n q converges towards αpf q " 1 2 pλpf q `1{λpf qq.

Step 3.-At this stage the coefficients ν n pf i q and the eigenvalues αpν n q are positive real numbers. Let U n be a small open neighborhood of ν n " pν n pf i qq in ∆ N . As a consequence of the first step, the map ν 1 P U n Þ Ñ w ν 1 contains a small neighborhood of w νn in Π Γ Ă NSpX; Rq. Thus, after a small perturbation of ν n we may assume that w νn is an element of R `NSpX ; Zq. According to Proposition 3.9 and Remark 3.11, when Γ contains parabolic elements, we may further choose w νn to be proportional to the pullback rπ 0 A n s of an ample class.

The equation satisfied by w νn is αpν n qw νn " ř i ν n pf i qf i pw νn q. Write w νn " η n wn for some wn in NSpX; Qq and η n in R `; the equation becomes αpν n q wn " N ÿ i"1 ν n pf i qf i p wn q.

(5.12) This is a linear relation of the form β 0 wn " ř i β i f i p wn q, where wn and the f i p wn q belong to NSpX; Qq and the β i are positive real numbers (with β 0 ą 1). Thus, given any ε ą 0, there is a relation of the form β0 wn " ř i βi f i p wn q where the coefficients βi are rational numbers which are ε-close to the original β i . This proves that we can perturb ν n one more time to add the assumption that the ν n pf i q and αpν n q are positive rational numbers. 5.4. Arithmetic equidistribution. Assume that the normal projective surface X and the subgroup Γ of AutpXq are defined over some number field k Ă Q. (X is not assumed to be smooth here.) For y in XpQq, let m y denote the uniform probability measure supported on the Galois orbit of y, m y " 1 degpyq ÿ y 1 PGalpQ:kqpyq δ y 1 ;

(5.13) here, degpyq is the degree of the closed point defined by y, or equivalently the cardinality of the orbit of y under the action of the Galois group GalpQ : kq, and the sum ranges over all points y 1 in this orbit. A sequence px j q of points of XpQq is generic if the only Zariski closed subset of X containing infinitely many of the x j 's is X. Equivalently, px j q converges to the generic point of X for the Zariski topology.

In the following theorem, Γ ν is the group generated by the support of ν, and S ν is the current given by Proposition 5.4 and Remark 5.5 (associated to the normalized class xw|wy ´1{2 w). Theorem 5.9. Let X be a normal projective surface defined over a number field k. Let ν be a probability measure on AutpX k q with finite support F and rational weights νpf q P Q `, for f in F . Assume that (i) P ν w " αw for some ample class w in NSpX; Qq and some eigenvalue α ą 1; (ii) px j q P XpQq N is a generic sequence such that each x j is a periodic point of Γ ν .

Then, the sequence of probability measures pm x j q converges towards the measure S ν Ŝν , and this measure is Γ ν -invariant.

It is important here that w is a rational class, that is w P NSpX; Qq instead of just NSpX; Rq, since we rely on results of Kawaguchi, Yuan and Zhang that require this assumption. It is also crucial that X is not supposed to be smooth because this result will be applied to the model X 0 constructed in §3.3. When Γ ν is non-elementary, the eigenvector w must be proportional to w ν and α " αpνq (as in Lemma 5.2).

As explained in Section 5.1, a consequence of the theorem is that the limit S ν ^Sν of the sequence pm x j q, depends only on Γ ν (but not on the weights νpf q); this will ultimately imply that Assumption (ii) happens only in very rare situations.

Example 5.10. Under the assumption of Theorem 5.9, assume furthermore that X is an abelian surface. Since Γ ν has a periodic point x 1 , the stabilizer Γ x 1 " Stab Γν px 1 q has finite index in Γ ν ; conjugating by a translation we can take x 1 as the neutral element for the group law on X » C 2 {Λ. Then, the periodic points of Γ x 1 (and of Γ ν ) are exactly the torsion points of X (see §4.1). By the equidistribution theorem of Szpiro, Ullmo, and Zhang, the measures m x j converge towards the Haar measure of X (see [START_REF] Szpiro | Équirépartition des petits points[END_REF]). Also, Γ x 1 is induced by a subgroup Γx 1 of GL 2 pCq preserving the lattice Λ, and Γ ν is a group of affine automorphisms with linear part given by Γx 1 and translation part given by the finite subset Γ ν p0q Ă X. Every cohomology class u in H 1,1 pX; Rq has a distinguished representative, given by the unique translation invariant p1, 1q-form ω u on X such that rω u s " u. Since Γ ν acts by affine automorphisms, the operator α ´1P ν preserves ω wν , and the current S ν is given by xw ν |w ν y ´1{2 ω wν . Thus, for abelian surfaces, Theorem 5.9 corresponds to the theorem of Szpiro, Ullmo, and Zhang together with the fact that ω wν ^ωwν is proportional to the volume form inducing the Haar measure on X.

Preliminary remarks for the proof of Theorem 5.9. Set Γ " Γ ν . Let π : Y Ñ X be a minimal resolution of X; it is unique, up to isomorphism (see [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]§III.6], Theorems (6.1) and (6.2)), and Γ lifts to a group of automorphisms Γ Y of Y ; we shall also consider ν as a measure on Γ Y . The pull-back π ˚: PicpXq Ñ PicpY q is an embedding, and an isometry for the intersection form. Since π ˚NSpX ; Rq is Γ Y -invariant and contains classes with positive self-intersection, we deduce that Γ is elementary if and only if Γ Y is, and Π Γ Y " π ˚ΠΓ Ă π ˚NSpX ; Rq if Γ is non-elementary. Also, π ˚w satisfies P ν pπ ˚wq " απ ˚w in NSpY ; Rq.

If Γ is elementary, we apply [15, Theorem 3.2] on Y . Two cases may occur. In the first case, Γ Y fixes a class u ‰ 0 in the closure of the positive cone PospY q; but then xπ ˚w|uy ą 0 because xπ ˚w|π ˚wy ą 0, and xP ν pπ ˚wq|uy " xπ ˚w|uy because u is invariant: this contradicts α ą 1. So, we are in fact in the second case: Γ Y (resp. Γ) contains a loxodromic element f and preserves the pair of lines RrT f s Y RrT f s. So, even when Γ is elementary, we know that it contains a loxodromic element. Now, assume that Pic 0 pXq is non-trivial; equivalently, Pic 0 pY q is non-trivial. Since Γ Y contains a loxodromic element, we deduce from [15, Theorem 10.1] that Y is a blow-up of an abelian surface (for Pic 0 pY q is trivial when Y is birationally equivalent to a rational, K3, or Enriques surfaces). But then, X is smooth and is also a blow-up of an abelian surface. If X itself is not an abelian surface, the exceptional divisor E of the blow up is Γ-invariant; as above since w is ample and α ą 1 we obtain a contradiction. So X is abelian, and Theorem 5.9 follows from the discussion in Example 5.10. Thus in what follows, we may assume Pic 0 pXq " 0 to simplify the exposition.

Proof of Theorem 5.9. We now assume that Pic 0 pXq " 0 and that Γ ν contains a loxodromic element.

The proof is based on standard ideas from arithmetic equidistribution theory. For the reader's convenience we provide background and details (see also [START_REF] Gyu | Equidistribution of periodic points of some automorphisms on K3 surfaces[END_REF] for the applicability of arithmetic equidistribution in this context). Changing w into a multiple, we may assume w P NSpX; Zq. Multiplying the equation ř f νpf qf ˚pwq " αw by the least common multiple b of the denominators, we obtain the linear relation ÿ f PF npf qf ˚pwq " dw;

( 5.14) in which d " bα and the coefficients npf q " bνpf q are positive integers such that pf ˚Lq bnpf q " L bd (5.16) up to an isomorphism of line bundles that we do not specify. From this identity, Kawaguchi constructs in [40, §1] a function ĥL : XpQq Ñ R `which satisfies the relation ř f npf q ĥL ˝f " d ĥL and differs from the naive Weil height function associated to L only by a bounded error. It will be referred to as as the canonical stationary height (associated to ν and L). This height function may be decomposed as a sum of continuous local height functions, see [40, §4]. Arakelov theory also provides a canonical adelic metric on pX, Lq; in particular, for each place v of k, there is a metric | ¨|v on the line bundle pX kv , L kv q, where k v is an algebraic closure of the v-adic completion of k, such that ź

ÿ f PF npf q " b ă d " bα (
f PF |spf pxqq| npf q v " |spxq| d v (5.17)
for every local section s of L defined over k. In our setting, an embedding k Ă C is fixed; it corresponds to one of the places of k. The adelic metric corresponding to that place gives a continuous metric on L, and from the relation (5.17) and the uniqueness of the current S ν we see that the curvature current of the metric is precisely the current S ν from Proposition 5.4 (see also Remark 5.5).

Lemma 5.11. A point x P XpQq satisfies ĥL pxq " 0 if and only if its Γ ν -orbit is finite.

Proof (see [START_REF] Kawaguchi | Canonical heights, invariant currents, and dynamical eigensystems of morphisms for line bundles[END_REF], Prop. 1.3.1). Let k 1 be any number field containing k. The set tx P Xpk 1 q ; ĥL pxq " 0u is Γ ν -invariant and by Northcott's theorem it is finite, so every element of that set has a finite orbit. Let us prove the other implication. Iterating the relation ř f npf q ĥL ˝f " d ĥL and evaluating it on a periodic point x yields α n ĥL pxq " ř gPΓ ν ‹n pgq ĥL pgpxqq where ν ‹n is the n-th convolution of ν. The right hand side is bounded because ĥL pgpxqq takes only finitely many values, and on the left hand side the term α n goes to `8; thus ĥL pxq " 0, as asserted.

L p ś vPM k B v q λ L pH 0 pX, Lq b A k {H 0 pX, Lqq (5.18)
does not depend on the choice of Haar measure. Taking tensor products, we get a sequence of adelic metrized line bundles pL bn q ně1 , and by definition the arithmetic volume of L is

x vol χ pX, Lq " lim sup nÑ`8 χpX, L bn q n 3 {6 . (5.19)
This is to be compared with the usual volume volpX, Lq of L, which by definition is the limsup of 2 n 2 h 0 pX, L bn q, as n tends to `8. A fundamental inequality of Zhang asserts that if px j q is a generic sequence in XpQq, lim inf j ĥL px j q ě

x vol χ pX, Lq 3volpX, Lq .

(5.20)

This follows from an adelic version of the Minkowski theorem on the existence of integer points in lattices (see e.g. [START_REF] Zhang | Small points and adelic metrics[END_REF] or Lemma 5.1 in [START_REF] Chambert | Mesures de Mahler et équidistribution logarithmique[END_REF]).

As for the usual volume, the arithmetic volume can be interpreted in terms of arithmetic intersection. Indeed, to L is associated an arithmetic degree y degpc 1 pLq 3 q, and it is shown in [START_REF] Zhang | Small points and adelic metrics[END_REF] that x vol χ pX, Lq " y degpc 1 pLq 3 q ě 0 (see also [START_REF] Kawaguchi | Canonical heights, invariant currents, and dynamical eigensystems of morphisms for line bundles[END_REF]Thm 2.3.1]). Thus, the existence of a generic sequence of periodic points px j q shows that x vol χ pX, Lq " 0 and ĥL px j q " x vol χ pX, Lq for all j. We are now in position to apply Yuan's equidistribution theorem (see [START_REF] Yuan | Big line bundles over arithmetic varieties[END_REF][START_REF] Berman | Growth of balls of holomorphic sections and energy at equilibrium[END_REF]): the sequence of measures pm x j q converges towards the probability measure S ν ^Sν as j goes to 8. If f is any element of Γ ν , the points f px j q also form a generic sequence of Γ-periodic points. Since the actions of Γ and GalpQ : kq commute, we infer that f ˚pm x j q " m f px j q , so taking the limit as j Ñ 8 yields f ˚pS ν ^Sν q " S ν ^Sν , and finally S ν ^Sν is Γ ν -invariant. 5.5. Density of active saddle periodic points. Let f be a loxodromic automorphism of X. We say that a periodic point of f is active if it is contained in the support of the measure of maximal entropy µ f . From [START_REF] Dujardin | Laminar currents and birational dynamics[END_REF][START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF] we know that a saddle periodic point that is not contained in any f -periodic curve is active (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF], Theorem 8.2). Theorem 5.12. Let X be a compact Kähler surface, and Γ be a non-elementary subgroup of AutpXq that contains a parabolic automorphism. Then, given any non-empty open subset V Ă X (for the Euclidean topology), there exists a point x P V and a loxodromic element f P Γ such that x is an active saddle periodic point of f . In particular, the union of the supports of the measures µ f , for f P Γ lox , is dense in X.

Before proceeding to the proof, let us point out the following fact, which readily follows from Lemma 3.5, together with the fact that an irreducible curve with negative self-intersection is determined by its class in NSpX; Zq Ă NSpX; Rq. Lemma 5.13. Let U and U 1 be two disjoint open subsets of PpNSpX; Rqq containing nef classes and introduce the set ApU, U 1 q " tf P AutpXq ; f is loxodromic, PprT f sq P U and PprT f sq P U 1 u.

Then, the union of all periodic curves of all elements of ApU, U 1 q is a finite set of curves.

Proof of Theorem 5.12. Pick g in Γ par . Since Γ is non-elementary we can conjugate g by an element of Γ lox to produce a pair g, h P Γ par with distinct fixed points BH X .

Step Assume that X is a blow-up of an abelian surface A, and pick f in Γ lox . By Lemma 4.3, its periodic points are dense, and all of them are active because µ f is the pull back to X of the Haar measure on A. Thus any open subset of X contains active saddle periodic points.

From now on, assume that X is not a blow-up of an abelian surface.

Step 2.-From Section 3.1, g preserves a unique fibration π g : X Ñ B g and the automorphism induced by g on B g is periodic. Replacing g by some iterate, we assume that π g ˝g " π g . Let U Ă B g be a small disk containing no critical value of π g . There is a real analytic diffeomorphism Φ : π ´1pU q Ñ U ˆR2 {Z 2 and a real analytic map ϕ : U Ñ R 2 such that π U ˝Φ " π g and g Φ :" Φ ˝g ˝Φ´1 satisfies g Φ pb, zq " pb, z `ϕpbqq (5.21) for all points pb, zq P U ˆR2 {Z 2 . According to [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Cantat | Dynamics of automorphism groups of complex projective surfaces: parabolic maps and invariant measures[END_REF], ϕ is generically of maximal rank: there is a finite set Z Ă U such that pDϕq b : T b U Ñ R 2 has rank 2 for every b P UzZ; hence, tb P U ; ϕpbq P Q 2 {Z 2 u is dense in U. If ϕpbq " pa 0 {N, b 0 {N q for some integers a 0 , b 0 and N , then every point q " pb, zq in the fiber is fixed by g N Φ and

pDg N Φ q x " ˆid 2 0 N pDϕq b id 2 ˙.
(5.22) Thus, in any holomorphic coordinate system px, yq in which π g expresses as π g px, yq "

x, the differential of g N at the fixed point Φ ´1pqq is of the form p 1 0 a 1 q with a ‰ 0.

Step 3.-The invariant fibrations π g and π h are transversal in the complement of a proper Zariski closed set Tangpπ g , π h q. According to Lemma 5.13 and Lemma 3.13, we can find an integer N ą 0, and a divisor F Ă X such that all elements g N ˝h N with ě 1 are loxodromic and do not have any periodic curve outside F .

Step 4.-Let D be the union of the singular and multiple fibers of π g and of π h , of Tangpπ g , π h q, and of the divisor F ; D is a divisor of X. Let V be an open subset of X. Then V contains a small ball V 1 such that -V 1 does not intersect D; -π g pV 1 q and π h pV 1 q are topological disks U g and U h in B g and B h respectively; -there are local coordinates px, yq in V 1 (resp. x in U g and y in U h ) such that pπ g q |V 1 px, yq " x and pπ h q |V 1 px, yq " y.

Step 2 provides a point px 0 , y 0 q P V 1 and an integer N ą 0 such that g N fixes the fiber of π g through px 0 , y 0 q pointwise, h N fixes the fiber of π h through px 0 , y 0 q pointwise, and pDg N q px 0 ,y 0 q " ˆ1 0 a 1 ˙, and pDh N q px 0 ,y 0 q " ˆ1 b 0 1

˙(5.23)
for some non-zero complex numbers a and b. If P Z is sufficiently large, f N " pDg N q px,yq ˝pDh N q px,yq is a loxodromic automorphism, px 0 , y 0 q is a fixed point of f N which is not contained in a periodic curve of f N (because px 0 , y 0 q is not in F ), which is shown to be a saddle by an explicit computation. Thus, as explained before the proof, px 0 , y 0 q is active, and we are done.

Measure rigidity and Kummer examples.

Theorem 5.14. Let X be a complex projective surface and let Γ be a subgroup of AutpXq. Assume that (i) X and Γ are defined over a number field k Ă Q;

(ii) Γ is non-elementary and contains a parabolic automorphism.

If Γ has a Zariski dense set of finite orbits, then every loxodromic automorphism in Γ is a Kummer example.

Proof.

Step 1.-From the Zariski dense set of finite orbits we can extract a generic sequence of Γ-periodic points px j q P XpQq N . Since Γ is non-elementary, it contains a loxodromic element f . The isolated periodic points of f are defined over Q, because X and f are defined over Q, and the non-isolated periodic points of f form a finite number of f -periodic curves (see §3.2). Thus, we can find a Zariski dense set of Γ-periodic points x 1 i in XpQq. If Z Ă X is an irreducible curve that contains infinitely many of the x 1 i , then Z is defined over Q too. There are only countably many curves defined over Q. Thus, by a diagonal argument, we find an infinite sequence of periodic points x j P XpQq such that px j q is generic.

In what follows, px j q denotes such a generic sequence of periodic points. Consider the contraction π 0 : X Ñ X 0 of the union D Γ of all Γ-periodic curves (see Proposition 3.9); the group Γ also acts on the normal projective surface X 0 . Note that the projection pπ 0 px j qq P X 0 pQq N is also generic.

Step 2.-There exists a Γ-invariant measure m such that µ f " m for all loxodromic f . Fix an arbitrary element f in Γ lox . By [18, Lemma 2.9], Π Γ is defined over Q so applying Proposition 5.8 we obtain a sequence of probability measures pν n q. Denote by S νn and S νn,0 the currents, on X and X 0 respectively, given by Proposition 5.4 and Remark 5.5; by construction π 0 S νn,0 " S νn , where the pull-back is obtained by locally pulling back the continuous potentials.

For the moment we fix the integer n. Theorem 5.9 shows that the sequence of probability measures m π 0 px j q converges towards S νn,0 ^Sνn,0 as j goes to `8. In particular, S νn,0 ^Sνn,0 is a fixed Γ-invariant probability measure µ 0 :" lim j m π 0 px j q that does not depend on n. Since S νn,0 has continuous potentials, this measure gives no mass to proper analytic subsets of X 0 . Let µ be the probability measure which is equal to π 0 pµ 0 q on XzD Γ and gives no mass to D Γ . Since S νn has continuous potentials, µ " S νn ^Sνn . In X, the sequence `mx j ˘converges to µ. Indeed, if a subsequence of `mx j ˘converges towards some probability measure λ, then pπ 0 q ˚λ " µ 0 , and since µ 0 does not charge any point of X 0 , we infer that λ is equal to π 0 pµ 0 q on XzD Γ and does not charge D Γ , which means that λ " µ. Thus, by compactness of the set of probability measures, m x j converges towards µ. Now, we let n Ñ 8. By Proposition 5.8 (5), Proposition 5.4, and Lemma 5.7, S νn ^Sνn " µ converges towards µ f as n goes to `8. Thus µ " µ f for all loxodromic elements f in Γ. In particular, µ is f -ergodic hence Γ-ergodic.

Step 3.-Conclusion.

As already explained, µ gives no mass to proper algebraic subsets of X. Furthermore, Theorem 5.12 implies that the support of µ is equal to X. Thus, Theorem 0.2 of [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF] (see also [START_REF] Cantat | Dynamics of automorphism groups of complex projective surfaces: parabolic maps and invariant measures[END_REF]) shows that µ is absolutely continuous with a smooth density. Since µ " µ f , the Main Theorem of [START_REF] Cantat | Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy[END_REF] implies that pX, f q is a Kummer example, as was to be shown.

5.7.

From Kummer examples to Kummer groups. In this paragraph we prove the following theorem, which together with Theorem 5.14 implies Theorem B. Its formulation, involving subgroups generated by unipotent parabolic elements, is intended for further use in §8.

Theorem 5.15. Let X be a compact Kähler surface, and Γ be a non-elementary subgroup of AutpXq containing parabolic elements. Assume that, given any pair of unipotent parabolic elements pg, hq P Γ 2 par , every loxodromic element f P xg, hy is a Kummer example. Then pX, Γq is a Kummer group.

Proof. Consider the birational morphism π 0 : X Ñ X 0 given in Proposition 3.9. By Proposition 3.12, there exists a loxodromic transformation f , of the form h ˝g for some unipotent elements g, h in Γ par , such that its maximal invariant curve D f coincides with D Γ . By assumption, f is a Kummer example, which entails that X 0 is a quotient A{G, with A " C 2 {Λ an abelian surface and G a finite subgroup of AutpAq generated by a diagonal map g 0 P GL 2 pCq of order 2, 3, 4, 5, 6 or 10 (see §4.3.2).

The group Γ induces a group of automorphisms of X 0 . View X 0 as an orbifold: its fundamental group is Λ ¸G and its universal cover Ă X 0 is C 2 . Concretely, this means that X 0 is the quotient of C 2 by the group of affine transformations with linear part in G and translation part in Λ. The canonical hermitian metric on C 2 is invariant under the affine action of Λ ¸G. If h : C 2 Ñ C 2 is a lift of some h P Γ to Ă X 0 ( 5 ), the norm of D hpx,yq with respect to this hermitian metric is constant along the orbits of Λ ¸G, hence it is bounded since the action is co-compact. This implies that the holomorphic map

px, yq P C 2 Þ Ñ D hpx,yq (5.24)
is constant. So, if we denote by r Γ the group of all possible lifts of all elements of Γ to C 2 " Ă X 0 , then r Γ is a group of affine transformations that contains Λ ¸G as a normal subgroup and satisfies Γ{pΛ ¸Gq " Γ. The action by conjugation of Γ on Λ ¸G preserves the subgroup Λ of translations. Therefore, Λ is also normal in r Γ: this shows that r Γ induces a group of automorphisms Γ A " r Γ{Λ of the abelian surface A " C 2 {Λ that covers X 0 . The proof is complete.

AROUND THEOREM B: CONSEQUENCES AND COMMENTS

6.1. Corollaries. The following corollary of Theorem B applies for instance to general Wehler examples defined over Q. Corollary 6.1. Let X be a smooth projective surface and let Γ be a subgroup of AutpXq. Assume that:

(i) X and Γ are defined over a number field; (ii) X is not an abelian surface; (iii) Γ contains a parabolic automorphism, and has no invariant curve.

Then Γ admits only finitely many finite orbits.

Proof. Suppose Γ has infinitely many finite orbits; since Γ does not preserve any curve, these orbits form a Zariski dense subset. Let g be a parabolic automorphism of Γ. If the fibration π g were Γ-invariant, then Γ would preserve the curve Ť yPΓpxq π ´1 g pπ g pyqq for every Γ-periodic point x. Thus, there is an element h in Γ that does not preserve π g , and h ´1 ˝g ˝h P Γ is a parabolic map associated to a different fibration. Hence Γ is non-elementary (see §3.1.3) and Theorem B shows that Γ is a Kummer group. But, 5

To prove the existence of such a lift, note that h maps the regular part of X 0 to itself, so first lift h| RegpX0q to C 2 zπ ´1pSingpX 0 qq, which is simply connected, and then use Hartogs extension to extend h accross the discrete set π ´1pSingpX 0 qq. since X is not abelian, a Kummer subgroup of AutpXq admits an invariant curve (see Lemma 4.7): this contradiction concludes the proof.

The next result is in the spirit of the "dynamical Manin-Mumford problem". Corollary 6.2. Let X be a smooth projective surface and Γ be a subgroup of AutpXq, both defined over a number field. Suppose that Γ is non-elementary and contains parabolic elements. Let C Ă X be an irreducible curve containing infinitely many periodic points of Γ. Then, (1) either C is Γ-periodic and is fixed pointwise by a finite index subgroup of Γ;

(2) or pX, Γq is a Kummer group and C comes from a translate of an abelian subvariety (of dimension 1).

In both cases the genus of C is 0 or 1. Thus, a curve of genus ě 2 contains at most finitely many periodic points of Γ.

To be specific, with the notation of §4.3, the second assertion means the following: there is a translate E `t of an elliptic curve E Ă A such that q X pCq " q A pE `tq. Moreover, if we choose the origin of A at a periodic point of Γ A , we can choose t to be a torsion point of A. We keep these notations in the following proof.

Proof. Let PerpCq be the set of periodic points of Γ in C; it is Zariski dense in C, for C is irreducible. The Zariski closure of ΓpPerpCqq is either a Γ-invariant curve or X.

In the first case C is contained in D Γ , a finite index subgroup Γ 1 Ă Γ preserves C, and the restriction Γ 1 |C has infinitely many periodic points in C. In this case C has (arithmetic) genus 0 or 1 by [START_REF] Diller | Invariant curves for birational surface maps[END_REF]Theorem 1.1]. A group of automorphisms of a curve with at least three periodic orbits is finite, because it admits a finite index subgroup fixing 3 points; thus, a finite index subgroup of Γ fixes C pointwise.

In the second case, Theorem B shows that pX, Γq is a Kummer group. Since C cannot be periodic, its image q X pCq Ă A{G is a non-trivial curve whose lift to A contains a Zariski dense subset of Γ A -periodic points. Choose one of these periodic points as the origin of A. By Proposition 4.1 and Remark 4.2, the Γ A -periodic points are exactly the torsion points of A, and conclusion (2) follows from Raynaud's theorem (formerly known as the Manin-Mumford conjecture) [START_REF] Raynaud | Courbes sur une variété abélienne et points de torsion[END_REF]. 6.2. Finitely generated groups. It turns out that Γ is often defined over a number field when X is. Proposition 6.3. Let X be a projective surface defined over a number field k. Assume that AutpXq contains a loxodromic element, and that X is not an abelian surface. Then any finitely generated subgroup of AutpXq is defined over a finite extension of k. Corollary 6.4. If X is a K3 or Enriques surface defined over a number field k, AutpXq is defined over a finite extension of k. Indeed, AutpXq is finitely generated in this case (see [START_REF] Sterk | Finiteness results for algebraic K3 surfaces[END_REF]).

Proof of the proposition. It is enough to show that any automorphism f P AutpXq is defined over a finite extension of k. Under our assumption, AutpXq ˚Ă GLpH ˚pX, Zqq is infinite, AutpXq 0 is trivial, and the homomorphism f P AutpXq Þ Ñ f ˚P GLpH ˚pX, Zqq has finite kernel (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]Theorem 10.1]); more precisely, if H is any ample divisor, the stabilizer of rHs is a finite subgroup AutpX; rHsq of AutpXq.

Fix a finite extension k 1 of k and a basis of NSpX; Zq given by classes of divisors D i which are defined over k 1 . Fix an ample divisor H defined over k 1 . By assumption X and the D i are defined by polynomial equations over k 1 , in some P N . Now, consider an automorphism f of X, defined by polynomial formulas with coefficients in some extension K of k 1 . Any field automorphism ϕ P GalpK : k 1 q conjugates f to an automorphism f ϕ of X: this defines a map ϕ P GalpK : k 1 q Þ Ñ f ϕ P AutpXq. On the other hand xpf ϕ q ˚rD i s|rD j sy " xf ˚rD i s|rD j sy for any pair pi, jq because the divisors D i are defined over k 1 ; thus, pf ϕ q ˚" f ˚on NSpX; Zq, and f ϕ ˝f ´1 belongs to the finite group AutpX; rHsq, so the set tf ϕ ; ϕ P GalpK : kqu is finite, and we are done. 6.3. Open problems. In the case of the affine plane A 2 , it follows from [START_REF] Dujardin | The dynamical Manin-Mumford problem for plane polynomial automorphisms[END_REF] that any non-elementary subgroup of AutpA 2 k q, for any number field k, has at most finitely many finite orbits (see [START_REF] Dujardin | The dynamical Manin-Mumford problem for plane polynomial automorphisms[END_REF] for the definition of "non-elementary" in this case). This motivates the following question: Question 6.5. Is Theorem B true without assuming the existence of a parabolic element in Γ?

To understand the difficulties behind Question 6.5, let us comment on three arguments that required the hypothesis Γ par ‰ H. First, it was used to show that Π Γ Ă NSpX; Rq is defined over Q and to construct the projective surface X 0 (which is then used in the construction of the canonical stationary height). The point is that in general the contraction of the divisor D Γ is a well-defined complex analytic surface, but it is not projective (see [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Example 2.10] and [19, §11]). We expect that this issue could be circumvented by applying more advanced techniques from Arakelov geometry. Second, Theorem 5.12 also relies on the existence of parabolic elements; the point was to show that all active periodic points of all loxodromic elements of Γ cannot be simultaneously contained in some real surface. For instance, it is unclear to us whether there can exist a real projective surface X R , with a non-elementary subgroup Γ Ă AutpX R q, such that all periodic points of all elements f P Γztidu are contained in the real part XpRq of X. Third, parabolic automorphisms are crucially used in the classification of Γ-invariant probability measures given in [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Cantat | Dynamics of automorphism groups of complex projective surfaces: parabolic maps and invariant measures[END_REF]. We expect that the techniques from [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF][START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] will soon lead to a complete classification of Γ-invariant probability measures, for any non-elementary group Γ Ă AutpXq. Such a classification would then open the way to an extension of Theorem B to all non-elementary groups (defined over a number field). Remark 6.6. In [42, Question 3.3], Kawaguchi formulates two interesting questions which are closely related to our main results as well as to Question 6.5.

(1) First, he asks whether two loxodromic automorphisms f and g of a complex projective surface X with a Zariski dense set of common periodic points automatically have the same periodic orbits. As it is formulated, the answer is no, because of Kummer examples: if we start with two loxodromic automorphisms of an abelian surface A fixing the origin and generating a non-elementary subgroup, then one can blow-up the origin, and the automorphisms lift to automorphisms with the same periodic orbits (coming from torsion points of A), except for their fixed points on the exceptional divisors, which do differ (see Lemma 4.7, Assertion (2)). So, his question needs to be modified by asking whether f and g have the same periodic points, except for finitely many of them. Taking these comments into consideration, Kawaguchi's second question can now be reformulated as: if two loxodromic automorphisms f and g of a complex projective surface X have a Zariski dense set of common periodic points, then is it true that either f m " g n for some m, n ě 1, or f and g generate a Kummer group? This seems harder than Question 6.5, because common periodic points do not directly provide common periodic orbits. A natural companion to the last question is: when do two loxodromic automorphisms have the same measure of maximal entropy?

One may also ask for effective bounds on the cardinality of a maximal finite Γinvariant subset of XpCq in terms of the data (compare [START_REF] Demarco | Common preperiodic points for quadratic polynomials[END_REF]). Proposition 2.8 says that such a bound should at least depend on the degrees of the generators of Γ.

Lastly, a natural question is whether the number field assumption in Theorem B is necessary at all: this is what the next section is about.

FROM NUMBER FIELDS TO C

In this section we show how a specialization argument allows to extend Corollary 6.1 beyond the number field case. A full generalization of Theorem B to complex coefficients would require further ideas (see §7.4 for a short discussion). For concreteness we first treat the case of Wehler surfaces and then explain the extra ingredients required to address the general case. 7.1. Wehler surfaces. We resume the notation from §2. The complete linear system |L| parameterizing Wehler surfaces is a projective space of dimension 26, which yields a moduli space of dimension 17 modulo the action of AutpP 1 q 3 . There is a dense, Zariski open subset W 0 Ă |L| such that if X P W 0 , then X is a smooth Wehler surface and for every 1 ď j ‰ k ď 3, π j,k : X Ñ P 1 ˆP1 is a finite morphism. Let Γ X be the group generated by the three involutions σ i .

Theorem 7.1. If X is a smooth Wehler surface for which the projections π j,k : X Ñ P 1 ˆP1 are finite maps, then Γ X admits only finitely many finite orbits.

For the proof we follow the approach of [32, §5 and Thm D] closely.

Proof. Let G " pZ{2Zq ‹ pZ{2Zq ‹ pZ{2Zq with generators a 1 , a 2 , a 3 and let χ : G Ñ AutpXq be the unique homomorphism such that χpa i q " σ i . By definition, Γ X " χpGq. Let c i denote the class of the curve X X tz i " C st u. The subspace Zc 1 ' Zc 2 ' Zc 3 of NSpX, Zq is invariant by χpGq ˚Ă GLpNSpX, Zqq and this representation does not depend on X P W 0 : the matrices of the involutions σ i " χpa i q ˚in the basis pc 1 , c 2 , c 3 q have constant integer coefficients (see e.g. [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Lem. 3.2]). Thus we can define G lox (resp. G par ) to be the set of elements h P G such that for any X P W 0 , χphq acts as a loxodromic (resp. parabolic) map on NSpX, Zq. Here, we implicitly use the fact that the type of h P AutpXq is the same as the type of h ˚in restriction to any h ˚-invariant subspace of H 1,1 pX; Rq on which the intersection form is not negative definite. In particular, the type of h coincides with the type of h ˚as an isometry of Vectpc 1 , c 2 , c 3 q.

Fix a system of affine coordinates px, y, zq and write the equations of Wehler surfaces as in Equation (2.1); this gives a system of homogeneous coordinates on |L|, and |L| can be considered as a projective space defined over Q. Then, endow |L| » P 26 pCq with the Q-Zariski topology. Fix X P W 0 , let b P P 26 (for "base point") denote the parameter corresponding to X, and S be the closure of tbu for this topology: this is a subvariety of P 26 defined over Q in which b is, by construction, a generic point. We put S 0 " S X W 0 , and we restrict the universal family X Ñ P 26 of Wehler surfaces to a family X S 0 Ñ S 0 , with a fiber preserving action of the group G. The fiber over s is denoted by X s and the natural homomorphism G Ñ AutpX s q by χ s ; thus, X coincides with X b . Lemma 7.2. For every s P S 0 pCq, (1) X s is a smooth K3 surface which does not contain any fiber of π i,j , i ‰ j P t1, 2, 3u;

(2) h P G belongs to G lox (resp. G par ) if and only if χ s phq is a loxodromic (resp. parabolic) element of AutpX s q; (3) χ s pGq is a non-elementary subgroup of AutpX s q without invariant curve.

Proof of Lemma 7.2. The first assertion follows from the results of §2.1 and the inclusion S 0 Ă W 0 . Likewise χ s pGq has no invariant curve by §2.2. The second assertion follows from our preliminary remarks on the definition of G lox and G par , and it also implies that χ s pGq is non-elementary.

Assume now by contradiction that Γ X admits infinitely many finite orbits. Then: Lemma 7.3. For every s P S 0 pCq, χ s pGq has infinitely many finite orbits.

This lemma concludes the proof of the theorem. Indeed pick s P S 0 pQq. By Lemma 7.2, χ s pGq is non-elementary, contains parabolic elements; and the Zariski closure of the set of finite orbits of χ s pGq coincides with X s , for otherwise it would be an invariant curve. Then, by Theorem B, pX s , χ s pGqq must be a Kummer group. But X s is a K3 surface and a Kummer group on a non-abelian surface admits an invariant curve, so that we get a contradiction with Lemma 7.2.(3).

Proof of Lemma 7.3. We first describe the set of finite Γ X -orbits as a countable union of subvarieties by arguing as in §2.4.1. Let G d be the intersection of the kernels of all homomorphisms from G to groups of order ď d!; it is a finite index subgroup of G. For any action of G, if the orbit of a point x has cardinality ď d, then x is fixed by G d . Conversely if x is fixed by G d , then its G-orbit is finite. Define a subvariety Z d of X by Z d " tx P X ; @g P G d , χpgqpxq " xu .

(

Finally put Z " Ť dě1 Z d . Then the Γ X -orbit of x P X is finite if and only if x P Z. We can now define a subvariety Z d of X S 0 which is the fibered analogue of Z d , namely Z d " tps, xq ; x P X s , @g P G d , χ s pgqpxq " xu , (

and put Z " Ť Z d . We let Z s (resp. Z d,s ) be the intersection of Z (resp. Z d ) with X s . Set f " a 3 a 2 a 1 P G. An explicit computation shows that f P G lox and the eigenvalues of χ s pf q ˚on Vectpc 1 , c 2 , c 3 q are ´1, λpf q " 9 `4? 5, and 1{λpf q. The eigenline corresponding to ´1 is R ¨pc 1 ´3c 2 `c3 q, its orthogonal complement in Vectpc 1 , c 2 , c 3 q is the plane Π χspf q , and this plane contains the class c 1 `2c 2 `c3 . This class is ample, because it is a convex combination, with positive coefficients, of the Chern classes c i of the line bundles π i pO P 1 p1qq, i " 1, 2, 3. Since any invariant curve must be orthogonal to Π χspf q , we deduce that χ s pf q has no invariant curve (for all s P S 0 ). Now assume by contradiction that there is a parameter t P S 0 pCq such that Z t is finite. Let P n be the set of fixed points of χpf n q, so that P " Ť n P n is the set of all periodic points of χpf q; likewise let P n and P be their respective fibered versions. Note that Z Ă P. For fixed n, let Y n be the (reduced) subvariety of X S 0 whose underlying set is Z X P n . More precisely the sequence of subvarieties Y m n :" Ť m d"1 Z d X P n is non-decreasing with m, so it stabilizes, and we define Y n " Y m n for m sufficiently large; its fibers will be denoted by Y n,s (Y n,s is the intersection of Y n with X s , it may be non-reduced). For the generic point b the cardinality of Y n,b tends to infinity with n. Now, the argument is identical to that of Lemma 5.3 and Theorem D in [START_REF] Dujardin | The dynamical Manin-Mumford problem for plane polynomial automorphisms[END_REF] ( 6 ). For x P Y n,s , its multiplicity multpx, Y n,s q as a point in Y n,s is equal to its multiplicity as a fixed point of χ s pf q n . Nakayama's lemma implies that the function

s Þ Ñ ÿ xPYn,s multpx, Y n,s q
is upper semicontinuous for the Zariski topology, hence ÿ

xPYn,t multpx, Y n,t q ě ÿ xPY n,b multpx, Y n,b q ÝÑ nÑ8 `8. (7.3)
On the other hand, Z t is a finite set, so there exists n 0 such that for all n ě 1, Y n,t Ă P n 0 ,t , and the theorem of Shub and Sullivan [START_REF] Shub | A remark on the Lefschetz fixed point formula for differentiable maps[END_REF] asserts that for every x P P n 0 ,t , the multiplicity of x as a fixed point of χ t pf q n is bounded as n Ñ 8. This contradicts (7.3) and concludes the proof.

7.2. Groups without invariant curve. Let us recall Theorem C.

Theorem 7.4. Let X be a compact Kähler surface and let Γ be a subgroup of AutpXq.

Assume that (i) X is not an abelian surface, and (ii) Γ contains a parabolic element and has no invariant curve. Then Γ admits only finitely many finite orbits.

Proof. The idea is of the proof is the same as that of Theorem 7.1, however new technicalities arise. As in Corollary 6.1, Γ is automatically non-elementary, so X is projective.

Arguing by contradiction, we suppose that Γ admits infinitely many finite orbits. Applying Theorem D, we fix f P Γ lox without invariant curve. We also fix a parabolic element g P Γ.

Step 1.-Geometry of X.-Since Γ is non-elementary, X is a blow-up of an abelian surface, of a K3 surface, of an Enriques surface, or of the projective plane (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]Thm. 10.1]). In the first three cases, there is a unique minimal model ϕ : X Ñ X, and the exceptional divisor of ϕ is AutpXq-invariant. Since Γ has no invariant curve, X is already equal to its minimal model X, and since by assumption X is not abelian, in this case X is a K3 or an Enriques surface.

Step 2.-Reduction to a finitely generated subgroup.-The group generated by f and g satisfies assumption (ii) and since it is contained in Γ it also admits infinitely many finite orbits. From now on, we replace Γ by xf, gy and assume Γ to be finitely generated.

Step 3.-Specialization formalism.-Embed X into a projective space P N C . Fix a finite set of divisors E j in X whose classes form a basis of NSpX; Zq, let H Ă X be a 6 Our setting is actually simpler since we are dealing with automorphisms on a projective surface rather than birational mappings, so the properness issue analyzed in [START_REF] Dujardin | The dynamical Manin-Mumford problem for plane polynomial automorphisms[END_REF] is not relevant here. hyperplane section, and let Ω be a non-trivial rational section of K b2 X , where K X is the canonical bundle. If X is a K3 or an Enriques surface, we assume that Ω is regular, hence does not vanish. Let R Ă C be the Q-subalgebra generated by the coefficients of a system of homogeneous equations for X, the E j , H, and Ω, and by the coefficients of the formulas defining a finite symmetric set of generators of Γ. (We shall actually further enlarge R in §7.3.2.)

Let K " FracpRq. It is the field of rational functions of some algebraic variety V , defined over Q. There is a dense, Zariski open subset S of V , which may be assumed to be an affine subset, such that all elements of R correspond to regular functions on S.

Note that in what follows, by Zariski topology we mean the Q-Zariski topology. Nevertheless, since we will use transcendental arguments, SpCq will also be considered as a complex analytic space endowed with its Euclidean topology.

By specialization, i.e. evaluation of the elements of R at s P S, we can view X Ă P N , Γ, the E j , H, and Ω as families over S; that is, there is a proper morphism π : X Ñ S endowed with a group of fiber preserving automorphisms r Γ, together with a (complex) base point b P S so that the fiber X b may be identified with X, and furthermore r

Γ b " Γ, E j,b " E j , H b " H, r Ω b " Ω, etc.
The point b P S may be thought of as the generic point of S (i.e. its closure for the Zariski topology is S) so in particular b is a regular point of S and S is smooth in a complex neighborhood of b. If X is a K3 or an Enriques surface, changing S into some Zariski dense affine open subset, we may assume that r Ω s does not vanish on any X s .

Step 4.-Types of automorphisms and invariant curves.-Lemma 7.5. There is a Zariski open subset S 1 Ă S such that:

(1) above S 1 pCq, the projection X Ñ S is a submersion; for s P S 1 pCq, X s is smooth and it is not an abelian surface; (2) for s P S 1 pCq, f s is loxodromic and there exists a Euclidean neighborhood B of b such that for s P B, f s admits no invariant curve; (3) for s P S 1 pCq, g s is parabolic.

Proof of (1). The surface X b is smooth, and by construction there is a Zariski dense open subset S 1 of S above which X Ñ S is a submersion, so the set of parameters s for which X s is singular is a proper Zariski closed set S 1 (we will further reduce S 1 finitely many times in the proof, keeping the same notation). For the second conclusion, we observe that there is a Zariski open subset on which π : X Ñ S is a submersion, hence by Ehresmann's lemma the fibers in this open subset are diffeomorphic to X. On the other hand a surface which is diffeomorphic to a complex torus and possesses a nonelementary group of automorphisms is automatically an abelian surface. Since X is not abelian, we conclude that the same is true for any fiber X s , s P S 1 .

Proof of [START_REF] Baragar | Canonical vector heights on algebraic K3 surfaces with Picard number two[END_REF]. The loxodromic nature of f s follows from the lower semi-continuity of the dynamical degree for birational mappings on surfaces (see [START_REF] Xie | Periodic points of birational transformations on projective surfaces[END_REF]Thm. 4.3]), which we apply here for the C-Zariski topology.

Let us show that f s has no invariant curve for s P S 1 close to b. Indeed, recall from Proposition 3.7 that there is a uniform bound on the degree of an invariant curve.

Here we compute the degree of a curve on X s (resp. of f s ) with respect to the normalized ample class xH s |H s y ´1{2 rH s s induced by the hyperplane section H s . If c X is as in Proposition 3.7, the inequality 1 2 rDivp r Ω s qs ď c X xH s |H s y ´1{2 rH s s is satisfied on a Zariski open set. Then by Bishop's theorem, if ps i q is a sequence of points converging to b such that f s i preserves a curve C i , we can extract a subsequence along which pC i q converges towards a curve C in X b (see [26, §16] for the relevant notions). This curve is f b -invariant, which contradicts our assumption on f . The proof of the the third assertion of Lemma 7.5 is a little tedious and will be postponed to Section 7.3 below.

Step 5.-Conclusion.-We pick a point t in S 1 pQq X B, and argue exactly as in the case of Wehler surfaces. Indeed observe first that the assumptions of Corollary 6.1 are satisfied at the parameter t. Next, since all periodic points of f t of a given period are isolated, we can apply to f t the strategy of the proof of Theorem 7.1, based on Nakayama's lemma and the theorem of Shub and Sullivan; it implies that Γ t has infinitely many periodic orbits on X t , thereby reaching the desired contradiction. 7.3. Proof of Lemma 7.5 [START_REF] Baragar | Canonical vector heights on K3 surfaces with Picard number three-an argument for nonexistence[END_REF]. By Step 1, X is a K3 surface, an Enriques surface, or a blow-up of the projective plane. By the lower semi-continuity of the dynamical degree, for every s P S 1 , g s is parabolic or elliptic, so we need to show that the set of parameters for which g s is elliptic is Zariski closed. 7.3.1. K3 and Enriques surfaces. Assume that X is a K3 (resp. an Enriques) surface. Above S 1 pCq, every fiber X s has the diffeomorphism type of X b , in particular it is simply connected and K Xs is trivial (resp. its fundamental group is Z{2Z and K b2 X S is trivial), so it is also a K3 (resp. an Enriques) surface, for K3 (resp. Enriques) are characterized by these properties (see [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]Chap. VI]). For such a surface, the group th P AutpX s q ; h ˚" id on H 2 pX s ; Zqu has at most 4 elements (see [START_REF] Mukai | Automorphisms of Enriques surfaces which act trivially on the cohomology groups[END_REF]). The second Betti number is fixed, equal to 22 (resp. 10), and if h ˚P GLpH 2 pX s ; Zqq has finite order, then its order divides some fixed integer k, because GL 22 pZq (resp. GL 10 pZq) contains a finite index, torsion free subgroup. Thus, g s is elliptic if and only if g 4k s " id. This implies that the set of parameters s for which g s is elliptic is Zariski closed and does not contain b, and we are done in this case. 7.3.2. Rational surfaces. Now, we assume that X is rational. This case is slightly more delicate because there exists automorphisms of P 2 of arbitrary large finite order.

Let π g : X Ñ B be the invariant fibration of g, with B " P 1 since X is rational. Changing g by some positive iterate, we may assume that the action of g in the base B is the identity. As explained in [START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF][START_REF] Cantat | Elements generating a proper normal subgroup of the cremona group[END_REF], π g comes from a Halphen pencil; in particular, there is a pencil of curves in P 2 , defined by some rational function ϕ : P 2 P 1 , and a birational morphism η : X Ñ P 2 that blows up the base points of this pencil (and possibly other points too), such that π g coincides with ϕ ˝η. The last blow-up which is necessary to resolve the indeterminacies of ϕ provides a curve which is transverse to the fibration and has negative self-intersection. So, there is an irreducible multi-section E of π g such that E 2 ă 0.

Let us add to our Q-algebra R the coefficients of the formulas defining π g , ϕ, η, E, etc. Reducing S 1 if necessary, we get a family of automorphisms g s preserving each fiber of a genus 1 fibration π g,s : X s Ñ P 1 , with an irreducible multisection E s of negative self-intersection. As for K3 and Enriques surfaces, the following lemma finishes the proof. Lemma 7.6. There is an integer ą 0 such that if s P S 1 and g s is elliptic, then g s " id.

Proof. Set m " xrEs|rF sy where F is any fiber of π g . Above S 1 the surfaces X s are pairwise diffeomorphic, so they have the same second Betti number and there is an integer k ą 0 such that ph ˚qk " id for every elliptic automorphism of X s , for every s P S 1 . Now, if g s is elliptic, then pg k s q ˚rE s s " rE s s and this implies g k s pE s q " E s . Since g s preserves every fiber, and E s intersects every fiber in at most m points, we deduce that g k¨m! s fixes a point in each fiber. But an automorphism of a curve of genus 1 which fixes a point has order at most 12, so g 12k¨m! s " id Xs , and we are done. 7.4. Discussion. It would be interesting to extend Theorem B in its general form beyond number fields, that is, without assuming that D Γ " H. Fix pf, gq P Γ lox ˆΓpar , as above. The main difficulty appears in the following situation: Γ fixes D Γ pointwise, and for every parameter s P SpQq, the alleged Zariski dense set of finite orbits of Γ specializes as a finite subset of X s which intersects pD Γ q s . In that case, the theorem of Shub and Sullivan does not apply directly because it only deals with isolated fixed points; so, a finer understanding of the Lefschetz fixed point formula is required. We believe that the tools introduced in [START_REF] Iwasaki | Periodic points for area-preserving birational maps of surfaces[END_REF] and in an unpublished chapter of Xie's thesis [START_REF] Xie | Dynamique algébrique des applications rationnelles de surfaces[END_REF] may lead to a solution of this problem. is a restricted canonical vector height on V can ˆXpkq if and only if either rEs is orthogonal to V can , or ϕ : Xpkq Ñ R is bounded and satisfies ϕpxqrEs " ϕpf pxqqf ˚rEs for all f P Γ.

In this situation, we shall say that h rEs,ϕ is derived from the height h can .

Proof. If rEs is orthogonal to V can , then h rEs,ϕ " h can on V can ˆXpkq and there is nothing to prove. Otherwise, we can fix a class D P V can such that xrEs|Dy ‰ 0. If h rEs,ϕ is a canonical vector height, then ϕ " xrEs|Dy ´1ph rEs,ϕ ´hcan qpD, ¨q is bounded, because h rEs,ϕ pD, ¨q and hpD, ¨q are Weil heights associated to the same divisor. Furthermore ϕ satisfies ϕpxqrEs " ϕpf pxqqf ˚rEs because h rEs,ϕ and h can are Γ-equivariant and xrEs|f ˚pDqy " xf ˚rEs|Dy for all f P Γ. The reverse implication is straightforward.

Lemma 8.2. Assume that Γ contains a loxodromic element. Let C Ă X be an irreducible Γ-periodic curve. If the class of C belongs to V can , or if V can contains a Γ-periodic class D such that OpDq |C is ample, then the restriction homomorphism

f P Stab Γ pCq Þ Ñ f |C has finite image.
Proof. If C is Γ-periodic then its self-intersection is negative, the restriction of O X p´Cq to C has positive degree, and pO X p´Cqq |C is therefore ample. So, it is enough to consider the case where V can contains a periodic class D such that OpDq |C is ample.

By changing Γ in a finite index subgroup, we may assume ΓpCq " C. If σ is an automorphism of C over k, it maps C to a Γ-invariant curve C σ . There are only finitely many Γ-invariant irreducible curves (the components of D Γ ), because Γ contains a loxodromic element (see §3.2). Thus, the orbit of C under the group of automorphisms of C over k is finite and C is defined over a number field. In particular, Cpkq is dense in CpCq.

Set Γ 1 " Stab Γ pDq, and Pick x 0 P Cpkq. Then h can pD, yq " h can pD, x 0 q for every y in Γ 1 px 0 q; since h can pD, ¨q is a Weil height for D, and OpDq |C is ample, Northcott's theorem implies that tx P Cpk 1 q ; h can pD, xq " h can pD, x 0 qu is finite for every number field k 1 ; thus, Γ 1 px 0 q is a finite set. Since Cpkq is infinite, we can argue as in the proof of Corollary 6.2 to deduce that Γ 1 |C is finite, as asserted. Lemma 8.3. Assume Pic 0 pXq " 0 and identify PicpX; Rq with NSpX; Rq.

(1) If V can contains a class with positive self-intersection, then it contains Π Γ .

(2) If V can contains Π Γ , and if C is an irreducible rational Γ-periodic curve, then, h can pD, xq " 0 for every D P Π Γ and x P Cpkq.

Proof. If V can contains a class in the positive cone it contains the limit set LimpΓq, hence also Π Γ (see [18, §2.3]); this proves the first assertion. For the second one, pick a probability measure ν on Γ with finite support, and assume that P ν pDq " αpνqD for some D in Π Γ and some αpνq P R. Then, ř f νpf qh can pD, f pxqq " αpνqh can pD, xq by equivariance and linearity. On the other hand, OpDq |C has degree 0, because xD|Cy " 0, and is therefore trivial because C is rational. So, h can pD, ¨q is bounded on Cpkq. Since αpνq ą 1, this implies that h can pD, xq " 0 for every x P Cpkq. To conclude, note that such eigenvectors D generate Π Γ when we vary ν (see §5.3). 8.2. From canonical vector heights to Kummer groups.

Theorem E. Let X be a smooth projective surface and Γ be a subgroup of AutpXq, both defined over a number field k. Suppose that (i) Γ is non-elementary and contains parabolic elements; (ii) there exists a canonical vector height h can for pXpkq, Γq on a Γ-invariant subspace of PicpX; Rq which contains a divisor with positive self-intersection.

Then pX, Γq is a Kummer group. If in addition h can is defined on PicpX; Rq ˆXpkq, then X is an abelian surface.

The smoothness of X is essential for the last conclusion to hold; for instance, if pX 0 , Γq is a singular Kummer example with no Γ-invariant curve, we shall see that the Néron-Tate height induces a canonical vector height on PicpX 0 ; Rq ˆX0 pkq.

The remainder of this subsection is devoted to the proof of Theorem E. Let us already observe that once pX, Γq is known to be a Kummer group, the last conclusion readily follows from Lemmas 4.7 and 8.2. So all we have to show is that pX, Γq is a Kummer group. 8.2.1. Reduction to Pic 0 pXq " 0. Suppose Pic 0 pXq ‰ t0u. Then, Γ being nonelementary, [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]Theorem 10.1] shows that X is either an abelian surface or a blowup of such a surface along a finite orbit of Γ, and by definition pX, Γq is a Kummer group.

So, from now on, we assume Pic 0 pXq " t0u and identify PicpX; Rq with NSpX; Rq.

8.2.2.

A key lemma. Assumption (ii) provides a canonical vector height h can for pX, Γq in restriction to Π Γ (see Lemma 8.3). Recall from [START_REF] Joseph | Rational points on K3 surfaces: a new canonical height[END_REF][START_REF] Kawaguchi | Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint[END_REF] that for every f P Γ lox there exist canonical heights h f , respectively associated to the classes θ f , such that h f pf pxqq " λpf qh f pxq and h f pf ´1pxqq " λpf qh f pxq. They satisfy:

h f ě 0 on Xpkq; -if D f denotes the maximal invariant curve of f then, for x P Xpkq, h f pxq hf pxq " 0 if and only if x is a periodic point or x P D f (see [41, §5]).

Furthermore any Weil height h associated to θ f such that hpf pxqq " λpf qhpxq coincides with h f : indeed k :" h ´hf is bounded because h and h f are Weil heights associated to the same class, so the relation kpf pxqq " λpf qkpxq forces it to be identically zero. Thus, the next lemma follows immediately from the defining Properties (a), (b), and (c) (see [START_REF] Kawaguchi | Canonical vector heights on K3 surfaces-a nonexistence result[END_REF]Prop. 3.4] or [3, §1]). Lemma 8.4. If Pic 0 pXq " t0u and if h can is a canonical vector height for pX, Γq in restriction to Π Γ , then h can pθ f , ¨q " h f p¨q for every f P Γ lox . Remark 8.5. If x belongs to the maximal invariant curve D Γ and c belongs to Π Γ then h can pc, xq " 0. Indeed for every f P Γ lox , D Γ Ă D f so h can pθ f , xq " h f pxq " 0, and since the classes pθ f q f PΓ lox span Π Γ , the result follows by linearity. This extends the second assertion of Lemma 8.3 to invariant curves which are not rational. Now, recall that the classes θ f are normalized by xθ f |rκ 0 sy " 1. Let us view H X and PpH X q as subsets of tu P H 1,1 pX; Rq ; xu|uy ą 0, xu|rκ 0 sy " 1u. Setting

r Π Γ " Π Γ X tx¨|rκ 0 sy " 1u , (8.2)
LimpΓq can now be viewed as a subset of r Π Γ which generates Π Γ as a vector space. The starting point of the proof of Theorem E is the following key lemma, inspired from the approach of Kawaguchi in [START_REF] Kawaguchi | Canonical vector heights on K3 surfaces-a nonexistence result[END_REF]. Lemma 8.6. In addition to the assumptions of Theorem E, suppose that (iii) there exists f P Γ lox such that rθ f , θ f sXIntpConvpLimpΓqqq ‰ H, where Convp¨q is the convex hull and Intp¨q stands for the interior relative to r Π Γ .

Then pX, Γq is a Kummer group.

Proof. Set d " dim r Π Γ . Replacing k by a finite extension, we may assume that the birational morphism π 0 : X Ñ X 0 constructed in Proposition 3.9 is defined over k; this morphism contracts the maximal Γ-invariant curve D Γ .

Let ν be a probability measure on Γ, whose support is finite and contains f as well as elements of Γ par . Let w ν be the eigenvector of the operator P ν for the eigenvalue αpνq given by Lemma 5.2. As in Proposition 5.8, we may assume that w ν is a rational class and is the pull-back of an ample class rA 0 s on X 0 ; by muliplying w ν by a positive integer, we also assume that w ν is an integral class.

Let L be the line bundle given by the class w ν , and ĥL be the associated canonical stationary height, as in the proof of Theorem 5.9. This is the unique Weil height such that ř h νphq ĥL ˝h " αpνq ĥL . By the linearity of the canonical vector height and the uniqueness of ĥL , we get ĥL p¨q " h can pw ν , ¨q.

Pick w " aθ f `bθ f in the interior of ConvpLimpΓqq, with a, b in R `and a `b " 1. Then by linearity and Lemma 8.4, h can pw, ¨q " ah f `bh f . Caratheodory's theorem provides a subset Λ of LimpΓq such that |Λ| " d `1 and w belongs to the interior of the simplex ConvpΛq. By the density of fixed points of loxodromic elements in LimpΓq, we may assume that Λ is made of classes θ g for g in a finite subset Λ Γ of Γ lox . If ε ą 0 is small enough, w ´εw ν stays in IntpConvpΛqq; so, there are positive coefficients β g , for g P Λ Γ , such that w ´εw ν " ř gPΛ Γ β g θ g . By the linearity of h can , we infer that ah f p¨q `bh f p¨q "

ÿ gPΛ Γ β g h g p¨q `εĥ L p¨q. (8.3)
Now, if x P Xpkq is f -periodic, then h f pxq " h f pxq " 0, and since ĥL and the h g pxq are non-negative, we deduce that ĥL pxq " 0. The line bundle L is the pull-back of an ample line bundle A 0 on X 0 ; thus, by Lemma 5.11, the Γ ν -orbit of π 0 pxq in X 0 is a finite set. Since f has a Zariski dense set of periodic points, Theorem B implies that pX, Γ ν q is a Kummer group.

Since we can further choose Γ ν to contain any a priori given finite subset of Γ, we conclude from Theorem 5.15 that pX, Γq is a Kummer group.

From this point, the proof of Theorem E is completed in two steps. We first deal with the case dim Π Γ ď 4 by directly checking the Assumption (iii) of Lemma 8.6. This covers general Wehler surfaces (which is the setting of [START_REF] Kawaguchi | Canonical vector heights on K3 surfaces-a nonexistence result[END_REF]) since dim Π Γ " 3 in this case. The general case is treated in a second stage by a dimension reduction argument. 8.2.3. Conclusion when dim Π Γ ď 4. Since Γ is non-elementary, dim Π Γ ě 3, so we need to consider the cases dim Π Γ " 3 and dim Π Γ " 4 .

For dim Π Γ " 3, i.e. d " dimp r Π Γ q " 2, the intersection of r Π Γ with the positive cone is the Klein model of the hyperbolic disk H 2 . If Assumption (iii) is not satisfied, then for every f P Γ lox , LimpΓq is entirely contained on one side of the geodesic rθ f , θ f s. Fix 4 points in LimpΓq Ă BH 2 » S 1 , labelled in circular order pp 1 , p 2 , p 3 , p 4 q. By Lemma 3.3 provides elements f and g in Γ lox such that pθ f , θ g , θ f , θ ǵ q is arbitrary close to pp 1 , p 2 , p 3 , p 4 q. Then rθ f , θ f s intersects rθ g , θ ǵ s transversally in the disk H 2 , so LimpΓq intersects both sides of rθ f , θ f s, a contradiction. Now assume dim Π Γ " 4, i.e. d " 3. Then ConvpLimpΓqq is a convex body in dimension 3. The conclusion relies on the following lemma (see below for a proof). Lemma 8.7. Let p 1 , . . . , p 5 be five points in general position in R 3 . Then there is a pair of indices i ‰ j such that any plane containing p i and p j separates the remaining points into two non-trivial sets. Now fix such a 5-tuple of points in LimpΓq and approximate the given pair pp i , p j q by pθ f , θ f q, for some f P Γ lox . Then, rθ f , θ f s intersects the interior of ConvpLimpΓqq, and Lemma 8.6 finishes the proof of Theorem E (when dimpΠ Γ q ď 4).

Proof of Lemma 8.7. Express p 5 as a barycenter of the remaining points: p 5 " Barppp 1 ; α 1 q, . . . , pp 4 ; α 4 qq.

(8.4)

The general position assumption means that the α i are non-zero. If all the coefficients α i are positive, p 5 lies in the interior of the tetrahedron Convtp 1 , . . . , p 4 u, so any plane containing p 4 and p 5 separates the vertices of the triangle Convtp 1 , p 2 , p 3 u into two nontrivial parts. Therefore, the pair of indices pi, jq " p4, 5q works in this case. If exactly one of the coefficients is negative, say α , then consider the pair p , 5q, and denote by i, j, k the remaining indices (i.e. ti, j, ku " t1, 2, 3, 4uzt u). The segment rp , p 5 s cuts the triangle Convtp i , p j , p k u in a point h of its relative interior. Thus, any plane containing p and p 5 also contains h, and separates tp i , p j , p k u non-trivially. The only remaining case is when there are three positive and two negative coefficients, say α 3 and α 4 ; then, p 1 is a barycenter of the remaining points with 3 positive and 1 negative coefficients, so we are done in this case too. Lemma 8.9. Let g 1 and g 2 be unipotent parabolic elements in AutpXq with distinct invariant fibrations. Then, Γ 0 :" xg 1 , g 2 y is non-elementary and dimpΠ Γ 0 q ď 4.

Proof. Since π g 1 ‰ π g 2 , Γ 0 is non-elementary (see §3.1.3). The subspace W :" Fixpg 1 q X Fixpg 2 q of NSpX; Rq is fixed pointwise by Γ 0 . Thus W K is Γ 0 -invariant, it contains Π Γ 0 (see [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Prop. 2.8]), and all we need to show is that dimpW K q ď 4. To see this, note that a unipotent Euclidean isometry is the identity, thus if g P O `p1, dq is parabolic and unipotent, the structure of parabolic isometries of H d (see [34, §I.5]) implies that Fixpg ˚q Ă R d`1 is a subspace of codimension 2, and we are done.

From Lemma 8.9 and Section 8.2.3, we deduce that pX, xg 1 , g 2 yq is a Kummer group for every pair of unipotent elements g 1 , g 2 P Γ par generating a non-elementary subgroup. Thus by Theorem 5.15, pX, Γq itself is a Kummer group, and Theorem E is established. 8.3. Canonical vector heights on abelian surfaces. In this section, A is an abelian surface, defined over some number field k and Γ Ă AutpA k q is non-elementary. Denote by h NT : PicpAq ˆApkq Ñ R the Néron-Tate height on A; it vanishes identically on the torsion part of PicpAq, so we may also consider it as a function on PicpA; Rq ˆApkq. When 0 P A has a finite Γ-orbit, h NT is a canonical vector height (see [START_REF] Hindry | Diophantine geometry[END_REF]Thm. B.5.6]).

Let h can be a restricted canonical vector height for pA k , Γq, defined on some Γinvariant subspace V can of PicpA; Rq. Our goal is to compare it to h NT .

By definition, a divisor D on A is symmetric if r´1s ˚D is linearly equivalent to D, where rms denotes multiplication by m; likewise it is antisymmetric if r´1s ˚D » ´D or equivalently if D P Pic 0 pAq (see [START_REF] Hindry | Diophantine geometry[END_REF]Prop. A.7.3.2]). If f P AutpAq fixes the origin, it commutes to r´1s, so that f ˚preserves symmetry and antisymmetry. Remark 8.10. Any class rDs P NSpAq can be lifted to a symmetric divisor class D P PicpAq, which is unique up to a 2-torsion element in Pic 0 pAq. Thus, D admits a unique symmetric lift in PicpA; Rq. By using such a lift it makes sense to consider also h NT p¨, ¨q (resp. h can p¨, ¨q) as a function on NSpA; Rq ˆApkq (resp. on the projection of V can in NSpA; Rq). This observation will be used repeatedly in the following.

Remark 8.11. The Picard number of any complex abelian surface satisfies ρpAq P t1, 2, 3, 4u. When AutpAq contains a non-elementary group Γ, we obtain 3 ď dim Π Γ ď ρpAq ď 4. Moreover, ρpAq " 4 if and only if A is isogenous to B ˆB, for some elliptic curve B with complex multiplication (see [7, ex.10 p.142]). Proposition 8.12. If V can contains Pic 0 pAq b Z R, then Γ has a finite orbit in Apkq.

Proof. In this proof it is enough to consider h can as a function on Pic 0 pAq ˆXpkq, by composing with the natural homomorphism Pic 0 pAq Ñ Pic 0 pA; Rq.

Step 1.-If D is an element of Pic 0 pAq, then for every f P Γ lox , every periodic point x of f satisfies h can pD, xq " 0.

Assume f q pxq " x for some q ě 1. The endomorphism f q ´id is an isogeny of A because f is loxodromic (see Section 4.2). Thus, its dual pf q q ˚´id is an isogeny of Pic 0 pAq and we can find E P Pic 0 pAq such that pf q q ˚E ´E " D. By equivariance h can ppf q q ˚E, xq " h can pE, xq, and then by linearity h can pD, xq " 0.

Step 2.-Let k 1 be a finite extension of k, and let P be a subset of Apk 1 q. If, for every D P Pic 0 pAq, the set th NT pD, xq ; x P P u Ă R is bounded, then P is finite.

To see this, consider the abelian group Apk 1 q; by the Mordell-Weil theorem, its rank is finite, so modulo torsion it is isomorphic to Z r for some r ě 0. Set W k 1 " Apk 1 q b Z R, a real vector space of dimension r. Let H be an ample symmetric divisor on A k , then h NT pH, ¨q determines a positive definite quadratic form on V ; let x¨|¨y H be the bilinear pairing associated to h NT pH, ¨q. If s is an element of Apkq, and t s P AutpA k q is the translation by s, then D s :" H ´ts H is an element of Pic 0 pAq and h NT pD s , ¨q induces an affine linear form Apk 1 q Ñ R; namely, h NT pD s , ¨q " ´2xs|¨y H . Since x¨|¨y H is positive definite on W k 1 (see [START_REF] Hindry | Diophantine geometry[END_REF]Prop. B.5.3]), one can find r elements s i P Apk 1 q such that the linear forms i :" xs i |¨y H constitute a basis of the dual of W k 1 . Our assumption says that each i pP q is a relatively compact subset of R; this implies that P is contained in a compact, hence finite, subset of the lattice Apk 1 q Ă V .

Step 3.-Γ has a finite orbit. Let f be a loxodromic element of Γ, and x be a fixed point of f . Its Γ-orbit is made of fixed points of conjugates of f . Note that Γpxq is contained in Apk 1 q for some finite extension of k. By the first step, h can vanishes on Pic 0 pAq ˆΓpxq. Since h can and h NT are Weil heights, |h can pD, ¨q ´hNT pD, ¨q| ď BpDq for each divisor class D P Pic 0 pAq, where BpDq ě 0 depends on D. Thus, |h NT pD, Γpxqq| ď BpDq for every D P Pic 0 pAq, and the second step implies that Γpxq is finite. Proposition 8.13. Assume that the neutral element has a finite Γ-orbit. Then h can coincides with the Néron-Tate height on the set of symmetric divisors whose numerical class belongs to Π Γ , -the set of antisymmetric divisors, whenever one of these sets is contained in V can .

In the following proofs, we denote by Π s Γ the subspace PicpA; Rq made of symmetric elements E P PicpA; Rq such that rEs P Π Γ .

Proof. Let Γ 0 ď Γ be the finite index subgroup fixing the origin. Let us show that h can " h NT on Π s Γ ˆApkq. For this, we use Remark 8.10, identify Π s Γ with Π Γ , and consider h can and h NT as functions on Π Γ ˆApkq. Now, if f P Γ 0,lox , we get h NT pθ f , ¨q " h can pθ f , ¨q because the difference is bounded on Apkq, and is multiplied by λpf q ą 1 under the action of f (as in Lemma 8.4). Since the classes θ f , for f P Γ lox , generate Π Γ , our claim is established.

Let us now deal with antisymmetric divisors. Identifying Pic 0 pA k q with the dual abelian variety A _ k of A, we have to show that h can coincides with h NT on A _ pk 1 q for every finite extension k 1 of k. By the Mordell-Weil theorem A _ pk 1 q is a finitely generated abelian group so W _ k 1 :" A _ pk 1 q b Z R (8.5) is a real vector space of dimension r, for some r ă `8. Consider the function Φ : pD, xq Þ Ñ h can pD, xq ´hNT pD, xq. When D is fixed, ΦpD, ¨q is bounded: |ΦpD, xq| ď BpDq for all x P Apkq. On the other hand when x is fixed, Φ x pDq :" ΦpD, xq defines a linear form Φ x : W _ k 1 Ñ R. Applying the previous boundedness property to f pxq, for f ranging in Γ 0 , and using the equivariance ΦpD, f pxqq " Φpf ˚D, xq we obtain that for every x P Apkq, Φ x is bounded on every Γ 0 -orbit Γ 0 pDq Ă W _ k 1 . We claim that this forces Φ x to vanish, which is the desired result. For this we analyze the dual action of Γ 0 . Let f be a loxodromic element of Γ 0 , and f _ k 1 be the induced linear map on W _ k 1 " A _ pk 1 q b Z R. Let L f be the linear lift of f to C 2 , as in §4.1 and 4.2. Lemma 8.14. The endomorphism f _ k 1 is semi-simple and its complex eigenvalues are complex conjugate to those of L f ; none of them has modulus 1.

Let us take this for granted and conclude the proof. Since f _ k 1 is semi-simple, W _ k 1 is a direct sum of f _ k 1 -invariant irreducible factors À W _ i , each of dimension 1 or 2. For each W _ i , denote by λ i the corresponding eigenvalue of f _ k 1 , and pick some D i P W _ i zt0u. If W _ i is a line, then λ i P R ˚and |λ i | ‰ 1. Since Φ x is bounded on tpf _ k 1 q n pD i q ; n P Zu, the line

W _ i is contained in ker Φ x . If W _ i is a plane, then f _ k 1 | W _
i is a similitude with |λ i | ‰ 1 and Argpλ i q ‰ 0 mod p2πZq.

If Φ x | W _ i ‰ 0, t|Φ x | ď BpD i quXW _
i is a strip, which furthermore contains the orbit tpf _ k 1 q n pD i q, n P Zu. This is not compatible with the properties of λ i , and this contradiction shows that W _ i Ă ker Φ x , so finally Φ x " 0, as claimed.

Proof of Lemma 8.14. The complex torus underlying A _ C is isomorphic to a quotient of the space of C-antilinear forms on C 2 . So, if f P AutpAq is induced by a linear map L f P GL 2 pCq, the automorphism of A _ determined by f ˚is induced by the conjugate transpose L t f (see [7, §2.4]). When f is loxodromic, the eigenvalues of L f satisfy |α| ă 1 ă |β|; we deduce that the automorphism of AutpA _ C q determined by f ˚is also loxodromic, with eigenvalues α and β, and the minimal polynomial of L t f is pX ´αqpX ´βq. Let P be the minimal, real, unitary polynomial such that pX ´αqpX ´βq divides P (by construction degpP q P t2, 3, 4u and P has no repeated factors). Since P pL t f q " 0, we infer that P pf _ k 1 q " 0 and the result follows.

Proposition 8.15. Let A k be an abelian surface defined over a number field k. Let Γ be a non-elementary subgroup of AutpA k q, for which the neutral element 0 P Apkq is periodic. Then one of the following situation occurs:

(1) NSpA, Rq " Π Γ and the Néron-Tate height is the unique canonical vector height on PicpA; Rq.

(2) NSpA, Rq " Π Γ K ' RrEs for some rEs P NSpA; Rqzt0u, and the canonical vector heights on PicpA; Rq are exactly the functions of the form h can pD, xq " h NT pD, xqx rEs|Dyϕpxq, where ϕ : Apkq Ñ R is any bounded function such that ϕpf pxqqf ˚rEs " ϕpxqrEs for all f in Γ.

Proof. When NSpA, Rq " Π Γ , Proposition 8.13 and the decomposition of any divisor class as a sum D " D s `Da with D s symmetric and D a antisymmetric imply that h can " h NT . So by Remark 8.11 we may assume that ρpAq " 4 and dimpΠ Γ q " 3. Pick rEs P Π K Γ zt0u. The line RrEs is Γ-invariant, and the intersection form is negative on RrEs; as a consequence, there is a homomorphism α rEs : Γ Ñ t`1, ´1u such that f ˚rEs " αpf qrEs for all f P Γ. Then for fixed x, Theorem E'. Let X be a smooth projective surface, defined over a number field k. Let Γ be a non-elementary subgroup of AutpX k q that contains parabolic elements. Let h can be a canonical vector height on PicpX; Rq ˆXpkq for the group Γ. Then, X is an abelian surface and h can is derived from a translate of the Néron-Tate height by a periodic point y of Γ: h can pD, xq " h NT pD, x `yq `xrEs|Dy ϕpxq for some almost-invariant class rEs P NSpX; Rq and some bounded function ϕ : Xpkq Ñ R such that ϕpf pxqqf ˚rEs " ϕpxqrEs for f P Γ.

∆
Note that h can is just a translate of h NT when E is numerically trivial. 8.4.2. Restricted canonical vector heights. Let us add the assumption Pic 0 pXq " 0, (

to the hypotheses of Theorem E. Our goal is to describe all possibilities for pV can , h can q. Since Pic 0 pXq " 0, X is not a blow-up of an abelian surface and Theorem E implies that pX, Γq is a Kummer group of type (2), (3), (4), or (5) in the nomenclature of §4.3.2. We make use of the notation of §4.3.1 and 4.3.2. The origin 0 P A is a fixed point of the cyclic group G, and the orbit Γ A p0q is finite. Since G is generated by a finite order homothety px, yq Þ Ñ pαx, αyq on A, G acts trivially on NSpA; Rq and on symmetric divisors. Thus, NSpA{G; Rq can be identified to NSpA; Rq and to the subspace of PicpA; Rq generated by symmetric divisors; let ι : NSpA; Rq Ñ NSpX; Rq (8.8) denote the corresponding embedding, given by ι " q X pq A q ˚. On the space of symmetric divisors, the Néron-Tate height is G-invariant and Γ-equivariant, so it induces a canonical vector height h A{G NT p¨, ¨q on A{G for Γ A . Then, it induces a restricted canonical vector height on ιpNSpA; Rqq ˆXpkq, namely h X NT : pD, xq Þ ÝÑ h A{G NT ppq X q ˚D, q X pxqq. (8.9) In what follows, we denote by E i the disjoint irreducible rational curves contracted by q X (see Lemma 4.7); their classes generate ιpNSpA; Rqq K Ă NSpX; Rq. The height h X NT vanishes on Ť i E i pkq, because the E i are mapped to torsion points of A.

Lemma 8.16. We have Π Γ " ιpΠ Γ A q Ă V can Ă ιpNSpA; Rqq.

Proof. The first equality comes from the equivariance of q X and q A . The first inclusion follows from Lemma 8.3 and the assumption (ii) of Theorem E. It remains to prove the last inclusion. If V can is not contained in q X pNSpA{G; Rqq, there is an index i, and a class D in Π K Γ X V can such that xD|E i y ą 0, i.e. OpDq| E i is ample. The action of Γ on Π K Γ factorizes through a finite group (see [18, Lem. 2.9]), so D is Γ-periodic and by Lemma 8.2, the group Γ| E i is finite; this contradicts Lemma 4.7, and the conclusion follows.

Let D be an element of Π Γ . By Lemma 8.3, h can pD, xq " 0 for all x P Ť i E i pkq. Thus, pD, xq Þ Ñ h can pιpDq, q ´1 X pq A pxqqq is a well defined restricted canonical vector height on Π Γ A ˆApkq (see Remark 8.10), which gives height 0 to the fixed points of elements of Gztidu. By Proposition 8.13, this height coincides with the Néron-Tate height on Π Γ A ˆApkq.

This yields a complete description of h can when V can " Π Γ . By Lemma 8.16 and Remark 8.11, the remaining possibility is that dimpV can q " 4 and dimpΠ Γ q " 3. Choose an almost Γ A -invariant class rEs in NSpA; Rq, as in Proposition 8.15, and a divisor F in X such that rF s " ιprEsq. Each element D P V can decomposes as a sum we get a new restricted canonical vector height on V can ˆXpkq that vanishes on Ť i E i pkq. This height comes from a canonical vector height on A{G, and since as seen before NSpA{G; Rq can be identified to NSpA; Rq, it yields a canonical vector height for pA, Γ A q restricted to the space of symmetric divisors. The second assertion of Proposition 8.15 entails that this last height is derived from the Néron-Tate height for some function ϕ; since Γ A contains G, and G fixes rEs, ϕ is G-invariant. Coming back to X, we get that h can is derived from the Néron-Tate height too. In formulas, h can pD, xq " h X NT pD, xq `xrF s|Dy Φpxq (8.13) given by w 0 Þ Ñ α ´2βw 0 (resp. w 1 Þ Ñ α ´1β 3 w 1 ). Since |α ´2β| ă 1 and |α ´1β 3 | ă 1, F has exactly two periodic points on each E i , namely two saddle fixed points.

2. 5 .

 5 Examples. 2.5.1. Consider the subgroup H of Z{2Z ‹ Z{2Z ‹ Z{2Z generated by f m 23 and f m 31

3. 4 .

 4 Proof of Theorem D. Let us first deal with the case where Γ is elementary. By [15, Theorem 3.2] there is a loxodromic element f P Γ such that pf ˚qZ has finite index in Γ ˚.

5 . 15 ) because α ą 1 .

 5151 Denote by D a divisor with class w, and by L the line bundle O X pDq. Since Pic 0 pXq is trivial, the Equality (5.14) implies â f PF

  Let A k denote the ring of adèles of the number field k. The sections of L defined over k determine a lattice H 0 pX, Lq in H 0 pX, Lq b A k , and the quotient pH 0 pX, Lq b A k q{H 0 pX, Lq is therefore compact. Denote by L the line bundle L endowed with its canonical adelic metric. For each place v, denote by B v Ă H 0 pX, Lq b k v the unit ball with respect to the v-adic component | ¨|v of the adelic metric of L. Let λ L be a Haar measure on H 0 pX, Lq b A k . The quantity χpX, Lq " log λ

( 2 )

 2 The second part of[START_REF] Kawaguchi | Canonical vector heights on K3 surfaces-a nonexistence result[END_REF] Question 3.3] asks whether two loxodromic automorphisms of a Wehler surface having a Zariski dense set of common periodic points automatically generate an elementary group. There are (singular) Kummer examples in the Wehler family (see [14, §8.2]), and they provide counter-examples to this question.

  in the closure of KahpXq). It is big because it is nef and has positive self-intersection (see e.g.[START_REF] Lazarsfeld | Positivity in algebraic geometry. I[END_REF] Thm 2.2.16]). Thus, (2) this vector w ν is the unique eigenvector of P ν in PospXq up to a positive scalar multiple; it is nef and big.

  Remark 8.8. The theorem of Steinitz (see[35, §13.1]) is a far-reaching generalization of Lemma 8.7. There is no analogue of this lemma in higher dimension (see[35, §4.7]), hence the need for a different argument when dim Π Γ ě 5.

8.2.4. Conclusion of the proof of Theorem E.

Recall from Lemma 3.1 that g ˚is virtually unipotent for every g P Γ par .

  x pDq " h can pD, xq ´hNT pD, xq, (8.6) defines a linear form on PicpA; Rq, which by Proposition 8.13 vanishes identically on Π s Γ . So, ∆pD, xq " xrEs|Dyϕpxq for some real valued function ϕ, and the conclusion follows from Lemma 8.1. 8.4. Synthesis. 8.4.1. Canonical vector heights. Putting together Theorem E and Proposition 8.15 gives:

  Define a function by setting ψpxq " xrF s|rF sy ´1h can prF s, xq on Ť i E i pkq and ψpxq " 0 otherwise. It satisfies the equivariance ψpf pxqqf ˚rF s " ψpxqrF s because h can is equivariant and rF s is almost invariant, and it is bounded because OpF q |E i is trivial for each E i . Now, if we set h 1 can pD, xq " h can pD, xq ´xrF s|Dy ψpxq (8.12)

		D " D 1 `xrF s|Dy xrF s|rF sy	rF s	(8.10)
	with D 1 in Π Γ . Then, for x in	Ť i E i pkq, we get
	h can pD, xq "	xrF s|Dy xrF s|rF sy	h can prF s, xq.	(8.11)

3.3. Γ-periodic curves, singular models and ampleness. Denote by Π K Γ the orthogonal complement of Π Γ with respect to the intersection form. Lemma 3.8. Let Γ Ă AutpXq be a non-elementary subgroup.(i) A curve C Ă X is Γ-periodic if and only if rCs P Π K Γ . (ii) If Γ contains a parabolic element, and C is irreducible, then C is Γ-periodic if andonly if C is contained in a fiber of π g for every g P Γ par .Proof. For (i), we note that since the intersection form is negative definite on Π K Γ , Γ acts on this space as a group of Euclidean isometries. Thus, if c P Π K Γ is an integral class, then Γ ˚pcq is a finite set. Since Π Γ is generated by nef classes, rCs belongs to Π K Γ if and only if each of its irreducible components does, so it is enough to prove the result for an irreducible curve. Now an irreducible curve C with negative self-intersection is uniquely determined by its class rCs; so if rCs is contained inΠ K Γ , we conclude that C is Γ-periodic. Conversely, if C is Γ-periodic, a finite index subgroup Γ 1 Ă Γ preserves C. If f P Γ 1lox , then xθ f | rCsy " 0 because f preserves the intersection form. But Vectpθ f , f P Γ 1 lox q is a Γ 1 -invariant subspace of Π Γ , hence by strong irreducibility

CANONICAL VECTOR HEIGHTS

Let k be a number field and k » Q be an algebraic closure of k. Let X be a projective surface defined over k and Γ be a subgroup of AutpX k q. We consider the vector space PicpX; Rq " PicpX k q b Z R (8.1) of R-divisors of X k modulo linear equivalence; doing so, we annihilate the torsion part of Pic 0 pXq. Keep in mind that when X is birational to an abelian variety, the vector space Pic 0 pX; Rq :" Pic 0 pXq b Z R is infinite-dimensional. Note that if PicpX k q is tensorized by Q instead of R and Property (a) is stated over Q we get an equivalent notion. Given any Γ-invariant subspace V Ă PicpX; Rq, one may also study the notion of restricted canonical vector height h : V ˆXpkq Ñ R. This is most significant when V contains classes with positive self-intersection, in which case it surjects onto Π Γ under the natural map D P PicpX; Rq Ñ rDs P NSpX; Rq. (In the following we use brackets to distinguish a class in NSpX; Rq from a class in PicpX; Rq.) If A k is an abelian variety and Γ is a subgroup of AutpA k q fixing its neutral element, the Néron-Tate height on A is a canonical vector height for Γ (see [START_REF] Hindry | Diophantine geometry[END_REF]Theorem B.5.6]). The same holds more generally if the neutral element is Γ-periodic, because in this case Γp0q is made of torsion points (see Remark 4.2).

In this section, we describe automorphism groups of surfaces which are non-elementary, contain parabolic elements, and possess a (restricted) canonical vector height h can : Theorems E, E', and E" show that pX, Γq is a Kummer group and h can is derived from the Néron-Tate height. 8.1. Invariant classes and canonical vector heights. In the following lemmas, h can is a restricted canonical vector height for pX, Γq, defined in some Γ-invariant subspace V can Ă PicpX; Rq. We shall say that a class rEs in NSpX; Rq is almost Γ-invariant if f ˚rEs " ˘rEs for all f in Γ. where Φ : Xpkq Ñ R is a bounded function which satisfies Φpf pxqqf ˚rF s " ΦpxqrF s for f P Γ. This function is equal to ψ on Ť i E i pkq and to ϕ ˝qX on its complement. To conclude, using the above notation, let us summarize these results in a (somewhat imprecise) statement.

Theorem E". Let X be a smooth projective surface, defined over a number field k, and such that Pic 0 pX k q " 0. Let Γ be a non-elementary subgroup of AutpX k q containing parabolic elements. Let h can be a restricted canonical vector height on V can ˆXpkq for the group Γ, where V can Ă PicpX; Rq is Γ-invariant and contains classes with positive self-intersection. Then pX, Γq is a Kummer group associated to an abelian surface A, V can is contained in ιpNSpA; Rqq and either V can " Π Γ and h can coincides with the Néron-Tate height h X NT ; -or Π Γ is a codimension 1 subspace of V can and h can is derived from h X NT .

APPENDIX A. KUMMER SURFACES OF TYPE [START_REF] Berman | Growth of balls of holomorphic sections and energy at equilibrium[END_REF] In case ( 6), we get a cyclic quotient singularity of type 1 5 p1, 2q, resolved by a string of two rational curves of respective self-intersections ´3 and ´2 (see [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]§III.6] and [44, §2]). The ring of invariant functions for the group G is generated by the four monomials u 0 " x 5 , u 1 " x 3 y, u 2 " xy 2 , u 3 " y 5 , and the quotient surface is locally isomorphic to the surface W in C 4 determined by the equations

, with coordinates pv 0 , w 0 q, pv 1 , w 1 q, and pv 2 , w 2 q respectively, and glue together the open sets V 0 ztw 0 " 0u and V 1 ztv 1 " 0u by v 1 " 1{w 0 , w 1 " v 0 w 3 0 , and the open sets

The result is a smooth surface Y , which is a neighborhood of a string of two smooth rational curves: the curve E 1 corresponding to the axis tp0, w 0 q ; w 0 P Cu (glued to tpv 1 , 0q ; v 1 P Cu) and the curve E 2 corresponding to the axis tp0, w 1 q ; w 1 P Cu (glued to tpv 2 , 0q ; v 2 P Cu); the self-intersections of these curves are respectively equal to ´3 and ´2. There is a G-invariant rational map from C 2 to Y , given in coordinates pv i , w i q by v 0 " x 5 , w 0 " y{x 2 ; v 1 " x 2 {y, w 1 " y 3 {x; v 2 " x{y 3 , w 2 " y 5 .

(A.1)

The surface Y is a desingularization of W " C 2 {G, with projection Y Ñ W given by

If F is a linear automorphism of C 2 that normalizes G and F induces a loxodromic automorphism of the torus C 2 {Λ 5 , then F px, yq " pαx, βyq for some eigenvalues α, β with |α| ą 1 ą |β|. On the quotient space W it acts by pu 0 , u 1 , u 2 , u 3 q Þ Ñ pα 5 u 0 , α 3 βu 1 , αβ 2 u 2 , β 5 u 3 q, and on Y it acts locally by pv 0 , w 0 q Þ Ñ pα 5 v 0 , α ´2βw 0 q (resp. pα 2 β ´1v 1 , α ´1β 3 w 1 q and pαβ ´3v 2 , β 5 w 2 q). In particular, the linear projective map induced by F on E 1 (resp. E 2 ) is