Parabolic Automorphisms of Projective Surfaces (after M. H. Gizatullin)

Grivaux, Julien

doi:10.17323/1609-4514-2016-16-2-275-298

Cited by 13 publications

(16 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following theorem was stated and proved in the present form by Cantat [Can01], and follows from a result of Gizatullin (see [Giz80], or [Gri16] for a survey); see also [DF01] for the birational case.…”

Section: The Case Of Surfacesmentioning

confidence: 76%

See 1 more Smart Citation

On the cohomological action of automorphisms of compact Kähler threefolds

Bianco¹

2017

Preprint

View full text Add to dashboard Cite

Extending well-known results on surfaces, we give bounds on the cohomological action of automorphisms of compact Kähler threefolds. More precisely, if the action is virtually unipotent we prove that the norm of (f n ) * grows at most as cn 4 ; in the general case, we give a description of the spectrum of f * , and bounds on the possible conjugates over Q of the dynamical degrees λ 1 (f ), λ 2 (f ). Examples on compact complex tori show the optimality of the results.Résumé. Nous Ãľtendons des rÃľsultats bien connus sur l'action en cohomologie des automorphismes des surfaces aux variÃľtÃľs compactes Kähler de dimension 3. Plus prÃľcisÃľment, si l'action est virtuellement unipotente nous montrons que la norme de (f n ) * croît au plus comme cn 4 ; dans le cas gÃľnÃľral, nous donnons une description du spectre de f * et des bornes sur les possibles conjuguÃľs sur Q des degrÃľs dynamiques λ 1 (f ), λ 2 (f ). Des exemples sur les tores complexes compacts montrent l'optimalitÃľ de ces rÃľsultats.An automorphism f : X → X of a compact Kähler manifold induces by pull-back of forms a linear automorphismwhich preserves the cohomology graduation, the Hodge decomposition, complex conjugation, wedge product and Poincaré duality.

show abstract

Section: The Case Of Surfacesmentioning

confidence: 76%

“…In each of the cases above, simple linear algebra arguments allow to further describe the situation. For the following result see for example [Gri16].…”

Section: The Case Of Surfacesmentioning

confidence: 99%

On the cohomological action of automorphisms of compact Kähler threefolds

Bianco¹

2017

Preprint

View full text Add to dashboard Cite

show abstract

“…Then for any t, the automorphism f t remains a parabolic isometry of H 2 (X t , Z). Therefore, thanks to the main result of [21] (see [24]), X t is a Halphen surface. Now thanks to Proposition 3.1, we can write X t as the blowup of nine points p i (t), 1 ≤ i ≤ 9 varying holomorphically with t. For t small enough, X t is anticanonical so the points p i (t) lie on a plane cubic curve C t .…”

Section: Julien Grivauxmentioning

confidence: 83%

Infinitesimal deformations of rational surface automorphisms

Grivaux¹

2017

Math. Z.

Self Cite

View full text Add to dashboard Cite

show abstract

“…There is a classification of genus 1 pencils of the plane up to birational conjugacy, which dates back to Halphen (see [19,21]): a Halphen pencil of index l is a pencil of curves of degree 2l with 9 base-points of multiplicity l. Every Halphen twist f preserves such a pencil; on X , the pencil corresponds to the genus 1 fibration which is g-invariant.…”

Section: Elliptic Elements Of Cr 2 (K)mentioning

confidence: 99%

Distortion in Cremona groups

Cantat,

Cornulier

2018

Preprint

View full text Add to dashboard Cite

We study distortion of elements in 2-dimensional Cremona groups over algebraically closed fields of characteristic zero. Namely, we obtain the following trichotomy: non-elliptic elements (i.e., those whose powers have unbounded degree) are undistorted; elliptic elements have a doubly exponential, or exponential distortion according to whether they are virtually unipotent.By definition, δ c,S (n) = ∞ if and only if c has finite order. Clearly, if S ⊂ T then δ c,S ≤ δ c,T . Also, δ c,S k (n) = δ c,S (kn). In particular, if S ⊂ T k , then δ c,S ≤ δ c,T (kn). If S T , it follows that δ c,S δ c,T , and if S ≃ T then δ c,S ≃ δ c,T .If S and T both generate G then S ≃ T and δ c,S ≃ δ c,T . Thus, when G is finitely generated, the ≃-equivalence class of the distorsion function only depends on (G, c), not on the finite generating subset; it is is called the distortion function of c in G, and is denoted δ G c , or simply δ c . The element c is called undistorted if δ c (n) n, and distorted otherwise.Example 1.1. Fix a pair of integers k, ℓ ≥ 2. In the Baumslag-Solitar groupIn the "double" Baumslag-Solitar group, one finds double exponential distorsion (see [22] and § 8).It is natural to consider distortion in groups which are not finitely generated. We say that an element c ∈ G is undistorted if δ H c (n) ≃ n for every finitely generated subgroup H of G containing c. Changing H may change the distorsion function δ H c ; for instance, if c is not a torsion element, it is undistorted in H = c Z but may be distorted in larger groups. Also, there are examples of pairs (G, c) such that c becomes more and more distorted, in larger and larger subgroups of G (see § 8). Thus, we have a good notion of distorsion, but the distorsion is not measured by an equivalence class of a function " δ G c ". We shall say that the distorsion type (or class) of c in G is at least f if there is finitely generated subgroup H containing c with f δ H c , and at most g if δ H c g for all finitely generated subgroup H containing c. If the distorsion type is at least f and at most f simultaneously, we shall say that f is the distorsion type of c. For instance, c may be exponentially, or doubly exponentially distorted in G.Example 1.2. Let k be a field. Let c be an element of the general linear group GL d (K); we have one of the following (see [25,24] and § 3)

show abstract

Parabolic Automorphisms of Projective Surfaces (after M. H. Gizatullin)

Cited by 13 publications

References 11 publications

On the cohomological action of automorphisms of compact Kähler threefolds

On the cohomological action of automorphisms of compact Kähler threefolds

Infinitesimal deformations of rational surface automorphisms

Distortion in Cremona groups

Contact Info

Product

Resources

About