arXiv subjects
Publications and source records attributed to Serge Cantat.
Ramanujam's theorem states that any connected finite-dimensional subgroup of the automorphism group $\mathrm{Aut}(X)$ of an irreducible variety $X$ is an algebraic group, in a natural way. In this note, we discuss the notion of dimension and extend Ramanujam's theorem to arbitrary (not necessarily irreducible) varieties.
Let $X$ be a variety of dimension $n$, and let $\mathrm{Aut}(X)$ be its automorphism group. When $X$ is quasi-affine, we prove that a solvable subgroup of $\mathrm{Aut}(X)$ that is generated by an irreducible family of automorphisms containing the identity is an algebraic subgroup. Our main applications concern arbitrary varieties. First, every connected solvable subgroup of $\mathrm{Aut}(X)$ is contained in a Borel subgroup and its derived length is $\leq n+1$. Second, the notion of solvable and unipotent radicals are well defined for any subgroup of $\mathrm{Aut}(X)$. Third, if $X$ is quasi-affine and connected and $\mathcal{B} \subset \mathrm{Aut}(X)$ is a Borel subgroup of derived length $n+1$, then $X$ is isomorphic to the affine $n$-space $\mathbb{A}^n$ and $\mathcal{B}$ is conjugate to the Jonquières subgroup.
We investigate the multiplier rigidity problem for polynomial automorphisms of $\mathbf{C}^2$. A first result states that a complex Hénon map of given degree is determined up to finitely many choices by its multiplier spectrum, or more generally by the unstable multipliers of its saddle periodic points. This is the counterpart in this setting of a classical result of McMullen for one-dimensional rational maps. For compositions of Hénon maps, the same rigidity holds provided the multi-degree and the multi-Jacobian are fixed. As in McMullen's theorem, this follows from the nonexistence of stable algebraic families in the corresponding parameter space. This in turn relies on precise asymptotic bounds for the Lyapunov exponents of the maximal entropy measure along diverging families.
Let $G$ be a compact Lie group. Let ${\mathsf{F}}_n$ be the free group of rank $n$. We describe the orbits of ${\mathsf{Aut}}(\mathsf{F}_n)$ on ${\mathsf{Hom}}(\mathsf{F}_n;G)$ when $n$ is sufficiently large. The dynamics stabilizes: orbit closures and invariant probability measures are algebraic, as in Ratner's theorems.
We proved that, in characteristic 0, if two dominant endomorphisms of the projective plane of degree at least 2 are conjugate by some birational transformation, then they are conjugate by an automorphism. We also gave counterexamples in positive characteristic.
We initiate the study of the ''algebraic growth'' of groups of automorphisms and birational transformations of algebraic varieties. Our main result concerns $\text{Bir}(\mathbb{P}^2)$, the Cremona group in $2$ variables. This group is the union, for all degrees $d\geq 1$, of the algebraic variety $\text{Bir}(\mathbb{P}^2)_d$ of birational transformations of the plane of degree $d$. Let $N_d$ denote the number of irreducible components of $\text{Bir}(\mathbb{P}^2)_d$. We describe the asymptotic growth of $N_d$ as $d$ goes to $+\infty$, showing that there are two constants $A$ and $B>0$ such that $$ A\sqrt{\ln(d)} \leq \ln \left(\ln \left(\sum_{e\leq d} N_e \right) \right) \leq B \sqrt{\ln(d)} $$ for all large enough degrees $d$. This growth type seems quite unusual and shows that computing the algebraic growth of $\text{Bir}(\mathbb{P}^2)$ is a challenging problem in general.
We study parabolic automorphisms of irreducible holomorphically symplectic manifolds with a lagrangian fibration. Such automorphisms are (possibly up to taking a power) fiberwise translations on smooth fibers, and their orbits in a general fiber are dense ([1]). We provide a simple proof that the associated Betti map is of maximal rank, in particular, the set of fibers where the induced translation is of finite order is dense as well. R{É}SUM{É}. Nous {é}tudions les automorphismes paraboliques des vari{é}t{é}s symplectiques holomorphes qui sont irr{é}ductibles et projectives.
We prove several new rigidity results for polynomial automorphisms of $\mathbb C^2$ with positive entropy. A first result is that a complex slice of the (forward or backward) Julia set is never a smooth, or even rectifiable, curve. We also show that such an automorphism cannot preserve a global holomorphic foliation, nor a real-analytic foliation with complex leaves. These results are used to show that under mild assumptions, two real-analytically conjugate automorphisms are polynomially conjugate. For mappings defined over a number field, we also study the fields of definition of multipliers of saddle periodic orbits.
We study orbit closures and stationary measures for groups of automorphisms of $p$-adic affine surfaces.
We first present an overview of our previous work on the dynamics of subgroups of automorphism groups of compact complex surfaces, together with a selection of open problems and new classification results. Then, we study two families of examples in depth: the first one comes from folding plane pentagons, and the second one is a family of groups introduced by Jérémy Blanc, which exhibits interesting new dynamical features.
Consider the four punctured sphere ${\mathbb{S}}_4^2$. Each choice of four traces, one for each puncture, determines a relative character variety for the representations of the fundamental group of ${\mathbb{S}}_4^2$ in ${\sf{SL}}_2(\mathbb{C})$. We classify the stationary probability measures for the action of the mapping class group ${\sf{Mod}}({\mathbb{S}}_4^2)$ on these character varieties.
We confirm a conjecture of Friedland and Milnor: if two polynomial automorphisms f and g in Aut(C^2) with dynamical degree >1 are conjugate by some holomorphic diffeomorphism ϕof C^2, then ϕis a polynomial automorphism; thus, f and g are conjugate inside Aut(C^2). We also discuss a number of variations on this result.
We study the hyperbolicity properties of the action of a non-elementary automorphism group on a compact complex surface, with an emphasis on K3 and Enriques surfaces. A first result is that when such a group contains parabolic elements, Zariski diffuse invariant measures automatically have non-zero Lyapunov exponents. In combination with our previous work, this leads to simple criteria for a uniform expansion property on the whole surface, for groups with and without parabolic elements. This, in turn, has strong consequences on the dynamics: description of orbit closures, equidistribution, ergodicity properties, etc. Along the way, we provide a reference discussion on uniform expansion of non-linear discrete group actions on compact (real) manifolds and the construction of Margulis functions under optimal moment conditions.
We initiate the study of random iteration of automorphisms of real and complex projective surfaces, or more generally compact K{ä}hler surfaces, focusing on the fundamental problem of classification of stationary measures. We show that, in a number of cases, such stationary measures are invariant, and provide criteria for uniqueness, smoothness and rigidity of invariant probability measures. This involves a variety of tools from complex and algebraic geometry, random products of matrices, non-uniform hyperbolicity, as well as recent results of Brown and Rodriguez Hertz on random iteration of surface diffeomorphisms.
In this paper we show that an affine space is determined by the abstract group structure of its group of regular automorphisms in the category of connected affine varieties. To prove this we study commutative subgroups of the group of automorphisms of affine varieties.
We classify invariant probability measures for non-elementary groups of automorphisms, on any compact Kähler surface X, under the assumption that the group contains a so-called "parabolic automorphism". We also prove that except in certain rigid situations known as Kummer examples, there are only finitely many invariant, ergodic, probability measures with a Zariski dense support. If X is a K3 or Enriques surface, and the group does not preserve any algebraic subset, this leads to a complete description of orbit closures.
We study distortion of elements in two-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 double exponential distortion when they are virtually unipotent and an exponential distortion otherwise.
In the first part of this article, we answer a question of I. Dolgachev, which is related to the following problem: given a birational map $f\in\mathrm{Bir}(\mathbb{P}^m_\mathbf{k})$ and a linear projective map $A\in\mathrm{PGL}_{m+1}(\mathbf{k})$, when is $A\circ f$ regularizable? Dolgachev's initial question is whether this may happen for all $A$ in $\mathrm{PGL}_{m+1}(\mathbf{k})$, and the answer is negative. We then look at the sequence $n\mapsto\mathrm{deg}\, f^n$, $f\in\mathrm{Bir}(\mathbb{P}^2_\mathbf{k})$. We show that there is no constraint on the sequence $n\mapsto\mathrm{deg}\, f^n-\mathrm{deg}\, f^{n-1}$ for small values of $n$. Finally we study the degree of pencils of curves which are invariant by a birational map. When f is a Halphen or Jonquières twist, we prove that this degree is bounded by a function of $\mathrm{deg}\, f$. We derive corollaries on the structure of conjugacy classes, and their properties with respect to the Zariski topology of $\mathrm{Bir}(\mathbb{P}^2_\mathbf{k})$.