An Example of a Domain with Non-Compact Automorphism Group

On a domain in C^2 with generic piecewise smooth Levi-flat boundary and non-compact automorphism group

In this paper, we prove that if D is a simply-connected domain in C^2 with generic piecewise smooth Levi-flat boundary and non-compact automorphism group, then D is biholomorphic to the bidisc. The proof is based on a careful analysis of invariant measures.

Pinchuk scaling method on domains with non-compact automorphism groups

In this paper, we characterize weakly pseudoconvex domains of finite type in $\mathbb C^n$ in terms of the boundary behavior of automorphism orbits by using the scaling method.

Examples of domains with non-compact automorphism groups

We give, in dimensions three or greater, an example of a bounded, pseudoconvex, circular domain in complex space with smooth real analytic boundary and non-compact automorphism group which is not biholomorphically equivalent to any Reinhardt domain. We give an analogous example in dimension two, but the domain fails to be smooth at one boundary point---indeed it is not in any Lipschitz class at the exceptional point.

On the boundary orbit accumulation set for a domain with non-compact automorphism group

For a smoothly bounded pseudoconvex domain $D\subset{\Bbb C}^n$ of finite type with non-compact holomorphic automorphism group $\text{Aut}(D)$, we show that the set $S(D)$ of all boundary accumulation points for $\text{Aut}(D)$ is a compact subset of $\partial D$ and, if $S(D)$ contains at least three points, it is connected and thus has the power of the continuum. We also discuss how $S(D)$ relates to other invariant subsets of $\partial D$ and show in particular that $S(D)$ is always a subset of the Šilov boundary.

Automorphism groups of compact complex supermanifolds

Let $\mathcal M$ be a compact complex supermanifold. We prove that the set $\mathrm{Aut}_{\bar 0}(\mathcal M)$ of automorphisms of $\mathcal M$ can be endowed with the structure of a complex Lie group acting holomorphically on $\mathcal M$, so that its Lie algebra is isomorphic to the Lie algebra of even holomorphic super vector fields on $\mathcal M$. Moreover, we prove the existence of a complex Lie supergroup $\mathrm{Aut}(\mathcal M)$ acting holomorphically on $\mathcal M$ and satisfying a universal property. Its underlying Lie group is $\mathrm{Aut}_{\bar 0}(\mathcal M)$ and its Lie super…

Bounded automorphism groups of compact complex surfaces

We classify compact complex surfaces whose groups of bimeromorphic selfmaps have bounded finite subgroups. We also prove that the stabilizer of a point in the automorphism group of a compact complex surface of zero Kodaira dimension, as well as the stabilizer of a point in the automorphism group of an arbitrary compact Kaehler manifold of non-negative Kodaira dimension, always has bounded finite subgroups.

Dynamics of automorphisms on compact Kähler manifolds

We study holomorphic automorphisms on compact Kähler manifolds having simple actions on the Hodge cohomology ring. We show for such automorphisms that the main dynamical Green currents admit complex laminar structures (woven currents) and the Green measure is the unique invariant probability measure of maximal entropy.

Exponential mixing for automorphisms on compact Kaehler manifolds

Let f be a holomorphic automorphism of positive entropy on a compact Kaehler surface. We show that the equilibrium measure of f is exponentially mixing. The proof uses some recent development on the pluripotential theory. The result also holds for automorphisms on compact Kaehler manifolds of higher dimension under a natural condition on their dynamical degrees.

Carleman approximation by holomorphic automorphisms of $\mathbb C^n$

We approximate smooth maps defined on non-compact totally real manifolds by holomorphic automorphisms of $\mathbb C^n$.

紧致Kähler流形上自同构的鞍型周期点的等分布

Equidistribution of saddle periodic points for automorphisms on compact Kähler manifolds

本文证明了紧致Kähler流形上具有简单上同调作用的全纯自同构的鞍型周期点关于平衡测度等分布，并推广到正拓扑熵的Kähler曲面情形。

Jordan property for groups of bimeromorphic automorphisms of compact Kähler threefolds

Let $X$ be a non-uniruled compact Kähler space of dimension 3. We show that the group of bimeromorphic automorphisms of $X$ is Jordan. More generally, the same result holds for any compact Kähler space admitting a quasi-minimal model.

Exponential mixing property for automorphisms of compact Kähler manifolds

Let $f$ be a holomorphic automorphism of a compact Kähler manifold. Assume moreover that $f$ admits a unique maximal dynamic degree $d_p$ with only one eigenvalue of maximal modulus. Let $μ$ be its equilibrium measure. In this paper, we prove that $μ$ is exponentially mixing for all d.s.h.\ test functions.

Automorphisms of $\mathbb C^k$ and associated compact complex manifolds

In this paper, we first construct $k$-dimensional compact complex manifolds from automorphisms of $\mathbb{C}^k$ which admit a fixed attracting point at infinity. Then, we charactize the fundamental group as well as the universal covering of the attracting basin of this fixed point thanks to a generalization of the method described by T. Bousch in his thesis.

CR manifolds with noncompact connected automorphism groups

The main result of this paper is that the identity component of the automorphism group of a compact, connected, strictly pseudoconvex CR manifold is compact unless the manifold is CR equivalent to the standard sphere. In dimensions greater than 3, it has been pointed out by D. Burns that this result follows from known results on biholomorphism groups of complex manifolds with boundary and the fact that any such CR manifold M can be realized as the boundary of an analytic variety. When M is 3-dimensional, Burns's proof breaks down because abstract CR 3-manifolds are generically not realizable a…

The Automorphism Group of a Domain with an Exponentially Flat Boundary Point

We study the automorphisms group action on a bounded domain in $\CC^n$ having a boundary point that is exponentially flat. Such a domain typically has a compact automorphism group. Our results enable us to generate many new examples.

紧 Kähler 流形自同构的所有阶指数混合与 CLT

Exponential mixing of all orders and CLT for automorphisms of compact K{ä}hler manifolds

本文研究上同调作用简单的紧 Kähler 流形自同构，证明其唯一最大熵测度对 Hölder 可观测量具有所有阶指数混合性，并推出 CLT，结果覆盖所有正熵紧 Kähler 曲面自同构。

广义Kausz紧化与完整共线空间的自同构群

The automorphism groups of generalized Kausz compactifications and spaces of complete collineations

本文研究了广义Kausz紧化与完整共线空间的自同构群，通过反canonical包的正性确定了其结构，部分结果基于先前的工作。

Super-potentials for currents on compact Kaehler manifolds and dynamics of automorphisms

We introduce a notion of super-potential (canonical function) associated to positive closed (p,p)-currents on compact Kaehler manifolds and we develop a calculus on such currents. One of the key points in our study is the use of deformations in the space of currents. As an application, we obtain several results on the dynamics of holomorphic automorphisms: regularity and uniqueness of the Green currents. We also get the regularity, the entropy, the ergodicity and the hyperbolicity of the equilibrium measures.

Tits alternative for automorphism groups of compact Kaehler manifolds

We survey some properties of the automorphism groups of compact Kaehler manifolds. In particular, we present recent results by Keum, Oguiso and Zhang on the structure of these groups from the Tits alternative point of view. Several other related results will be also discussed. This paper is written for the VIASM Annual Meeting 2012.

Characterization of ${\bf C}^n$ by its automorphism group

We show that if the group of holomorphic automorphisms of a connected Stein manifold $M$ is isomorphic to that of ${\bf C}^n$ as a topological group equipped with the compact-open topology, then $M$ is biholomorphically equivalent to ${\bf C}^n$.

容许紧的全纯自同构可迁群的紧连通复流形

Compact, connected, complex manifolds that admit a compact transitive group of holomorphic automorphisms

本文提出一种仅基于李理论标准结论的李代数方法，为紧复齐性流形相关经典结论给出新证明，还提供了将典范阿贝尔李代数与紧李代数上可积复结构关联的方式，拓展了前人研究。

A characterization of domains in $\mathbf C^2$ with noncompact automorphism group

Let $D$ be a bounded domain in $\mathbf C^2$ with a non-compact group of holomorphic automorphisms. Model domains for $D$ are obtained under the hypothesis that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type.

自守变换固定点集的有限性

Finiteness of the fixed point sets of automorphisms

研究有界域在复数空间中的自守变换固定点集大小，证明离散性迫使有限性，并获得紧子群中自守变换固定点数的统一界。

紧复曲面的自同构群：T-若尔当性质、Tits二择性与可解性

Automorphism groups of compact complex surfaces: T-Jordan property, Tits alternative and solvability

本研究针对紧复曲面的双解析自同构群，证明其挠子群具有几乎幂零性，并探究了该群的Tits二择性与几乎可解子群的几乎导出长度，推进了复曲面自同构群结构的相关研究。

作用于紧复可平行化流形的有限群

Finite groups acting on compact complex parallelizable manifolds

本文证明紧复可平行化流形的自同构群具有Jordan性质，并证明复李群中余紧格的外自同构群的有限子群具有有界阶。

Metacyclic groups as automorphism groups of compact Riemann surfaces

Let $X$ be a compact Riemann surface of genus $g\geq 2$, and let $G$ be a subgroup of $Aut(X)$. We show that if the Sylow $2$-subgroups of $G$ are cyclic, then $|G|\leq 30(g-1)$. If all Sylow subgroups of $G$ are cyclic, then, with two exceptions, $|G|\leq 10(g-1)$. More generally, if $G$ is metacyclic, then, with one exception, $|G|\leq 12(g-1)$. Each of these bounds is attained for infinitely many values of $g$.

On the exponent of the automorphism group of a compact Riemann surface

Let $X$ be a compact Riemann surface of genus $g\geq 2$, and let $Aut(X)$ be its group of automorphims. We show that the exponent of $Aut(X)$ is bounded by $42(g-1)$. We also determine explicitly the infinitely many values of $g$ for which this bound is reached and the corresponding groups. Finally we discuss related questions for subgroups $G$ of $Aut(X)$ that are subject to additional conditions, for example being solvable.

对具有大自同构群的紧复曲面上的双曲遍历平稳测度进行分类

Classifying Hyperbolic Ergodic Stationary Measures on Compact Complex Surfaces with Large Automorphism Groups

本文在不假设群含抛物元素的前提下，研究紧复曲面自同构群上有限支撑概率测度所生成的非初等群，并对双曲遍历 $μ$-平稳概率测度进行分类。

Model Theory of Compact Complex Manifolds with an Automorphism

Motivated by possible applications to meromorphic dynamics, and generalising known properties of difference-closed fields, this paper studies the theory CCMA of compact complex manifolds with a generic automorphism. It is shown that while CCMA does admit geometric elimination of imaginaries, it cannot eliminate imaginaries outright: a counterexample to 3-uniqueness in CCM is exhibited. Finite-dimensional types are investigated and it is shown, following the approach of Pillay and Ziegler, that the canonical base property holds in CCMA. As a consequence the Zilber dichotomy is deduced: finite-d…

Green currents for holmorphic automorphisms of compact Kahler manifolds

Let f be a holomorphic automorphism of a compact Kahler manifold (X,ω) of dimension k>1. We study the convex cones of positive closed (p,p)-currents T_p, which satisfy a functional relation $f^*(T_p)=λT_p, λ>1,$ and some regularity condition (PB). Under appropriate assumptions on dynamical degrees we introduce closed finite dimensional cones, not reduced to zero, of such currents. In particular, when the topological entropy h(f) of f is positive, then for some m, there is a positive closed (m,m)-current T_m which satisfies the relation $f^* T_m=\exp(h(f)) T_m$. Moreover,every quasi-p.s.h. func…

Several types of solvable groups as automorphism groups of compact Riemann surfaces

Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $Aut(X)$ be its group of automorphisms and $G\subseteq Aut(X)$ a subgroup. Sharp upper bounds for $|G|$ in terms of $g$ are known if $G$ belongs to certain classes of groups, e.g. solvable, supersolvable, nilpotent, metabelian, metacyclic, abelian, cyclic. We refine these results by finding similar bounds for groups of odd order that are of these types. We also add more types of solvable groups to that long list by establishing the optimal bounds for, among others, groups of order $p^m q^n$. Moreover, we show that Zomorrodian's bound…

The automorphism group and limit set of a bounded domain II: the convex case

For convex domains with $C^{1,ε}$ boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the automorphism group has finitely many components and the connected component of the identity is the almost direct product of a compact group and a non-compact connected simple Lie group with real rank one and finite center. In this case, we also show the limit set is homeomorphic to a sphere and prove a gap theorem: either the domain is biholomorphic to the unit ball (and the…

Bimeromorphic automorphisms groups of certain conic bundles

We study the groups of biholomorphic and bimeromorphic automorphisms of conic bundles over certain compact complex manifolds of algebraic dimension zero.

Zero entropy automorphisms of compact Kähler manifolds and dynamical filtrations

We study zero entropy automorphisms of a compact Kähler manifold $X$. Our goal is to bring to light some new structures of the action on the cohomology of $X$, in terms of the so-called dynamical filtrations on $H^{1,1}(X, {\mathbb R})$. Based on these filtrations, we obtain the first general upper bound on the polynomial growth of the iterations $(g^m)^* \, {\circlearrowleft} \, H^2(X, {\mathbb C})$ where $g$ is a zero entropy automorphism, in terms of ${\rm dim} \, X$ only. We also give an upper bound for the (essential) derived length $\ell_{\rm ess}(G, X)$ for every zero entropy subgroup $…

Explicit Examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy

We present the first explicit examples of a rational threefold and a Calabi-Yau threefold, admitting biregular automorphisms of positive entropy not preserving any dominant rational maps to lower positive dimensional varieties. The most crucial part is the rationality of the quotient threefold of a 3-dimensional torus of product type.

Basins of attraction of automorphisms in C^3

In this paper we shall give examples of maps and automorphisms with regions of attraction that are not simply connected.

Some dynamical properties of pseudo-automorphisms in dimension 3

Let $X$ be a compact Kähler manifold of dimension 3 and let $f:X\rightarrow X$ be a pseudo-automorphism. Under the mild condition that $λ_1(f)^2>λ_2(f)$, we prove the existence of invariant positive closed $(1,1)$ and $(2,2)$ currents, and we also discuss the (still open) problem of intersection of such currents. We prove a weak equidistribution result (which is essentially known in the literature) for Green $(1,1)$ currents of meromorphic selfmaps, not necessarily 1-algebraic stable, of a compact Kähler manifold of arbitrary dimension; and discuss how a stronger equidistribution result may be…

Dynamics of non cohomologically hyperbolic automorphisms of $\mathbb{C}^3$

We study the dynamics of a family of non cohomologically hyperbolic automorphisms $f$ of $\mathbb{C}^3$. We construct a compactification $X$ of $\mathbb{C}^3$ where their extensions are algebraically stable. We finally construct canonical invariant closed positive $(1,1)$-currents for $f^*$, $f_*$ and we study several of their properties. Moreover, we study the well defined current $T_f \wedge T_{f^{-1}}$ and the dynamics of $f$ on its support. Then we construct an invariant positive measure $T_f \wedge T_{f^{-1}}\wedge ϕ_{\infty}$, where $ϕ_{\infty}$ is a function defined on the support of $T…

Lie group structures on automorphism groups of real-analytic CR manifolds

Given any real-analytic CR manifold M, we provide general conditions on M guaranteeing that the group of all its global real-analytic CR automorphisms is a Lie group (in an appropriate topology). Our conditions are in particular satisfied when M is an arbitrary compact real-analytic hypersurface embedded in some Stein manifold.