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Serge Cantat

Publications and source records attributed to Serge Cantat.

At least 37 records · Page 2Linked to original sources
arXiv2020

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.

math.AG↗
arXiv2020

Automorphisms of compact Kähler manifolds with slow dynamics

We study the automorphisms of compact Kähler manifolds having slow dynamics. By adapting Gromov's classical argument, we give an upper bound on the polynomial entropy and study its possible values in dimensions $2$ and $3$. We prove that every automorphism with sublinear derivative growth is an isometry ; a counter-example is given in the $C^{\infty}$ context, answering negatively a question of Artigue, Carrasco-Olivera and Monteverde on polynomial entropy. Finally, we classify minimal automorphisms in dimension $2$ and prove they exist only on tori. We conjecture that this is true for any dimension.

math.DS↗
arXiv2020

Elements generating a proper normal subgroup of the Cremona group

Consider an algebraically closed field k and the Cremona group of all birational transformations of the projective plane over k. We characterize infinite order elements of this group having a non-zero power generating a proper normal subgroup of the Cremona group.

math.GR↗
arXiv2019

Commensurating actions of birational groups and groups of pseudo-automorphisms

Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes, for example a discrete countable group with Kazhdan Property (T), is birationally conjugate to a group acting by pseudo-automorphisms on some non-empty Zariski-open subset. We apply this argument to classify groups of birational transformations of surfaces with this fixed point property up to birational conjugacy.

math.AG↗
arXiv2019

On degrees of birational mappings

We prove that the degrees of the iterates ${\rm deg}(f^n)$ of a birational map satisfy $\liminf({\rm deg}(f^n))<+\infty$ if and only if the sequence ${\rm deg}(f^n)$ is bounded, and that the growth of ${\rm deg}(f^n)$ can not be arbitrarily slow, unless ${\rm deg}(f^n)$ is bounded.

math.AG↗
arXiv2015

Dynamical degrees of birational transformations of projective surfaces

The dynamical degree $λ(f)$ of a birational transformation $f$ measures the exponential growth rate of the degree of the formulae that define the $n$-th iterate of $f$. We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of $λ(f)$ and the structure of the conjugacy class of $f$. For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed and well ordered set of algebraic numbers.

math.AG↗
arXiv2013

Pseudo-automorphisms with no invariant foliation

We construct an example of a birational transformation of a rational threefold for which the first and second dynamical degrees coincide and are $>1$, but which does not preserve any holomorphic (singular) foliation. In particular, this provides a negative answer to a question of Guedj. On our way, we develop several techniques to study foliations which are invariant under birational transformations.

math.DS↗
arXiv2013

Constraints on automorphism groups of higher dimensional manifolds

In this note, we prove, for instance, that the automorphism group of a rational manifold X which is obtained from CP^k by a finite sequence of blow-ups along smooth centers of dimension at most r with k>2r+2 has finite image in GL(H^*(X,Z)). In particular, every holomorphic automorphism $f:X\to X$ has zero topological entropy.

math.CV↗
arXiv2013

Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups

Thanks to the theory of Coxeter groups, we produce the first family of Calabi-Yau manifolds $X$ of arbitrary dimension $n$, for which $\Bir(X)$ is infinite and the Kawamata-Morrison movable cone conjecture is satisfied. For this family, the movable cone is explicitly described; it's fractal nature is related to limit sets of Kleinian groups and to the Apollonian Gasket. Then, we produce explicit examples of (biregular) automorphisms with positive entropy on some Calabi-Yau manifolds.

math.AG↗
arXiv2012

Normal subgroups in the Cremona group (long version)

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane P^2 over k is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory, and algebraic geometry to produce elements in the Cremona group that generate non trivial normal subgroups.

math.AG↗
arXiv2012

Rational surfaces with a large group of automorphisms

We classify rational surfaces for which the image of the automorphisms group in the group of linear transformations of the Picard group is the largest possible. This answers a question raised by Arthur Coble in 1928, and can be rephrased in terms of periodic orbits of a birational action of an infinite Coxeter group on ordered point sets in the projective plane modulo projective equivalence. We also outline the classification of non-rational surfaces with large automorphism groups.

math.AG↗
arXiv2010

Holomorphic actions, Kummer examples, and Zimmer Program

We classify compact Kähler manifolds $M$ of dimension $n\geq 3$ on which acts a lattice of an almost simple real Lie group of rank $\geq n-1$. This provides a new line in the so-called Zimmer program, and characterizes certain type of complex tori by a property of their automorphisms groups.

math.AG↗
arXiv2009

Dynamique p-adique (d'après des exposés de Jean-Christophe Yoccoz)

This paper is an introduction to the dynamics of rational fractions over a p-adic field. It follows the lectures that Jean-Christophe Yoccoz gave during the États de la recherche in May 2006, and develops certain points. Most of the results to be explained here are due to Benedetto, Bézivin, Hsia, Lubin, Morton, Rivera-Letelier and Silverman. ----- Ce texte est une introduction à la dynamique des fractions rationnelles sur un corps $p$-adique. Il reprend le cours que Jean-Christophe Yoccoz avait fait lors des \emph{États de la recherche} de mai 2006 et en développe certains points. La plupart des résultats présentés sont dûs à Benedetto, Bézivin, Hsia, Lubin, Morton, Rivera-Letelier et Silverman.

math.NT↗