Projective Surface Automorphisms of Positive Topological Entropy from an Arithmetic Viewpoint

Kawaguchi, Shu

doi:10.1353/ajm.2008.0008

Cited by 32 publications

(45 citation statements)

References 21 publications

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“…By the description above, we have the following (see Kawaguchi [18] Proposition 2.5; use also the Hodge index theorem for (3) -(4)): Lemma 2.9. Let X, g be as in 2.6.…”

Section: Conventions and Preliminary Resultsmentioning

confidence: 99%

Automorphism groups and anti-pluricanonical curves

Zhang

2008

Mathematical Research Letters

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Abstract. We show the existence of an anti-pluricanonical curve on every smooth projective rational surface X which has an infinite group G of automorphisms of either null entropy or of type Z⋉Z, provided that the pair (X, G) is minimal. This was conjectured by Curtis T. McMullen (2005) and further traced back to Marat Gizatullin and Brian Harbourne (1987). We also prove (perhaps) the strongest form of the famous Tits alternative theorem.

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“…By the description above, we have the following (see Kawaguchi [18] Proposition 2.5; use also the Hodge index theorem for (3) -(4)): Lemma 2.9. Let X, g be as in 2.6.…”

Section: Conventions and Preliminary Resultsmentioning

confidence: 99%

Automorphism groups and anti-pluricanonical curves

Zhang

2008

Mathematical Research Letters

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“…This result has already been used, and proved, by the author and by Kawaguchi when M is a surface (see [6,4,14]), and by Zhang when f is an automorphism of a compact Kähler manifold with positive entropy (see [18]). We shall extend (a weak form of) Theorem A to the case of meromorphic transformations in section 3, Theorem B.…”

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confidence: 85%

Invariant Hypersurfaces in holomorphic Dynamics

Cantat

2010

Mathematical Research Letters

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ABSTRACT. We prove the following result, which is analogous to two theorems, one due to Kodaira and Krasnov and another one due to Jouanolou and Ghys. Let M be a compact complex manifold and f a dominant endomorphism of M. If there exist k totally invariant irreducible hypersurfaces W i ⊂ M, with k > dim(M) + h 1,1 (M) then f preserves a nontrivial meromorphic fibration. We then study the case where f is a meromorphic map.1. Introduction 1.1. Let M be a connected compact complex manifold of dimension n, and f : M → M be a surjective endomorphism of M. Let W ⊂ M be a hypersurface. By definition, W is totally invariant if f −1 (W ) = W. This property implies, and is stronger than, the forward invariance f (W ) = W. These notions coincide when f is an automorphism. The topological degree of f is then equal to the product of the multiplicity m and the topological degree of f |W : W → W ; the divisor defined by the jacobian determinant of f contains W with multiplicity m − 1.Theorem A. Let M be a compact complex manifold, and f be a dominant endomorphism of M. If there are k totally invariant hypersurfaces W i ⊂ M, with, then there is a non constant meromorphic function Φ and a non zero complex number α suchThis result has already been used, and proved, by the author and by Kawaguchi when M is a surface (see [6,4,14]), and by Zhang when f is an automorphism of a compact Kähler manifold with positive entropy (see [18]). We shall extend (a weak form of) Theorem A to the case of meromorphic transformations in section 3, Theorem B. Remarks.Received by the editors May 28, 2010. M is the sheaf of holomorphic 1-forms, and H i (M, Ω 1 M ) denotes the i-th Cech cohomology group of this sheaf. By Hodge theory, these cohomology groups are finite dimensional complex vector spaces (see [12], chapter 0, section 6, page 100). 1.2.2.The proof of Theorem A provides a slightly stronger statement: The number k in inequality ( * ) can be replaced by the total number of irreducible components of the W i . 1.2.3.If g is an endomorphism of the complex projective space P n (C) with two totally invariant hypersurfaces, then there is a constant α = 0 and a non constant meromorphic function Ψ such that ψ • g = αΨ deg(g) . This conclusion is different from the invariance propertyThe coordinate axis of the plane P 2 (C) are totally invariant under the action of the endomorphism g[x : y : z] = [x 2 : y 2 : z 2 ]. In this case,P 2 (C) )) = 2 + 1 = 3 is equal to the number of totally invariant lines, and all meromorphic functions Φ such that Φ • g = αΦ for some α ∈ C are indeed constant. The same example, but in dimension n, shows that Theorem A is sharp for M = P n (C). 1.2.4.Theorem A is analogous, in a dynamical setting, to the following statement: If a compact complex manifold M has k irreducible hypersurfaces, withthen there is a non constant meromorphic function on M (see [1], section IV.6, page 129, or [10,15] and [3,16]). Another similar statement has been obtained by Jouanolou and Ghys for foliated manifolds: If a codimension one (singular...

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“…The conjecture is known in many cases; see [35,Remark 1.8] for a comprehensive list including abelian varieties [25,46], automorphisms of smooth projective surfaces [23,24], as well as certain product varieties [43]. Recently the conjecture was proved for all regular endomorphisms of smooth projective surfaces [35]; the proof in the case of surfaces relies heavily on the birational classification.…”

Section: Introductionmentioning

confidence: 99%

“…The key to proving Theorem 1.2 is to construct a canonical height function associated to f , following a strategy developed by Silverman [45] and Kawaguchi [23] in dimension 2. Along the way, we obtain a hyper-Kähler analog of a result of Cantat and Kawaguchi [9,Proposition 4.1], the latter having been shown for surfaces.…”

Section: Introductionmentioning

confidence: 99%

Canonical Heights on Hyper-Kähler Varieties and the Kawaguchi–Silverman Conjecture

Lesieutre

Satriano

2019

International Mathematics Research Notices

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The Kawaguchi-Silverman conjecture predicts that if f : X X is a dominant rational-self map of a projective variety over Q, and P is a Q-point of X with Zariskidense orbit, then the dynamical and arithmetic degrees of f coincide: λ 1 (f ) = α f (P ). We prove this conjecture in several higher-dimensional settings, including all endomorphisms of non-uniruled smooth projective threefolds with degree larger than 1, and all endomorphisms of hyper-Kähler varieties in any dimension. In the latter case, we construct a canonical height function associated to any automorphism f : X → X of a hyper-Kähler variety defined over Q.

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Projective Surface Automorphisms of Positive Topological Entropy from an Arithmetic Viewpoint

Cited by 32 publications

References 21 publications

Automorphism groups and anti-pluricanonical curves

Automorphism groups and anti-pluricanonical curves

Invariant Hypersurfaces in holomorphic Dynamics

Canonical Heights on Hyper-Kähler Varieties and the Kawaguchi–Silverman Conjecture

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