Realizing Connected Lie Groups As Automorphism Groups Of Complex Manifolds

Complete hyperbolic Stein manifolds with prescribed automorphism groups

It is well-known that the automorphism group of a hyperbolic manifold is a Lie group.Conversely, it is interesting to see whether or not any Lie group could be prescribed asthe automorphism group of certain complex manifold. Whenthe Lie group $G$ is compact and connected, this problem has been completelysolved by Bedford-Dadok and independently by Saerens-Zame on 1987. Theyhave constructed \spc bounded domains $Ω$ such that $Aut(Ω)=G$. For Bedford-Dadok's $Ω, 0\le dim_{\Bbb C}Ω- dim_{\Bbb R}G\le 1$; for generic Saerens-Zame's$Ω,dim_{\Bbb C}Ω\gg dim_{\Bbb R}G$.J. Winkelmann has answered affirma…

On the group of holomorphic automorphisms of model surfaces

It is proved that the group of holomorphic automorphisms of holomorphically homogeneous nondegenerate (finite Bloom-Graham type + holomorphic nondegenaracy) model surface Q is a subgroup of the group of birational automorphisms of the ambient space (Cremona group) with uniformly bounded degree. An estimate of the degree of the automorphisms in terms of the dimension of the ambient space is given (Theorem 4). It is proved that none of the conditions of the theorem can be omitted. We consider also the question about connectedness of the automorphism group of Q (Theorem 7).

对称双圆盘上的等距映射，II

Isometries in the symmetrized bidisc, II

该研究证明了保持Kobayashi距离的从单位圆盘D到对称双圆盘G₂的映射f必为全纯或反全纯，完善了对称双圆盘上等距映射的相关理论。

Regularization of Local CR-Automorphisms of Real-Analytic CR-Manifolds

Let M be a connected generic real-analytic CR-submanifold of a finite-dimensional complex vector space E. Suppose that for every point a in M the Lie algebra hol(M,a) of germs of all infinitesimal real-analytic CR-automorphisms of M at a is finite-dimensional and its complexification contains all constant vector fields and the Euler vector field. Under these assumptions we show that: (I) every hol(M,a) consists of polynomial vector fields, hence coincides with the Lie algebra hol(M) of all infinitesimal real-analytic CR-automorphisms of M; (II) every local real-analytic CR-automorphism of M ex…

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。

Homogeneous Kobayashi-hyperbolic manifolds with high-dimensional group of holomorphic automorphisms

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-2$. This result complements an existing classification for automorphism group dimension $n^2-1$ and greater obtained without the homogeneity assumption.

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…

Homogeneous Kobayashi-hyperbolic manifolds with automorphism group of subcritical dimension

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-3$. This result complements existing classifications for automorphism group dimension $n^2-2$ (which is in some sense critical) and greater.

Further steps towards classifying homogeneous Kobayashi-hyperbolic manifolds with high-dimensional automorphism group

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 4$ whose group of holomorphic automorphisms has dimension either $n^2-4$, or $n^2-5$, or $n^2-6$. This paper continues a series of articles that achieve classifications for automorphism group dimension $n^2-3$ and greater.

可实现为有界域自同构群的非线性李群

Non-linear Lie groups that can be realized as automorphism groups of bounded domains

本文研究李群能否成为 C^n 有界域双全纯自同构群，在1990年连通线性李群结论基础上，给出满足该实现性质的非线性李群例子。

On the automorphism groups of hyperbolic manifolds

We show that there does not exist a Kobayashi hyperbolic complex manifold of dimension $n\ne 3$, whose group of holomorphic automorphisms has dimension $n^2+1$ and that, if a 3-dimensional connected hyperbolic complex manifold has automorphism group of dimension 10, then it is holomorphically equivalent to the Siegel space. These results complement earlier theorems of the authors on the possible dimensions of automorphism groups of domains in comlex space. The paper also contains a proof of our earlier result on characterizing $n$-dimensional hyperbolic complex manifolds with automorphism grou…

On automorphisms of blowups of $\mathbb{P}^3$

Let $π:X\rightarrow \mathbb{P}^3$ be a finite composition of blowups along smooth centers. We show that for "almost all" of such $X$, if $f\in Aut(X)$ then its first and second dynamical degrees are the same. We also construct many examples of finite blowups $X\rightarrow \mathbb{P}^3$, whose automorphism group $Aut(X)$ has only finitely many connected components. We also present a heuristic argument showing that for a "generic" compact Kähler manifold $X$ of dimension $\geq 3$, the automorphism group $Aut(X)$ has only finitely many connected components.

The Jordan property for Lie groups and automorphism groups of complex spaces

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic groups (not necessarily affine) over fields of characteristic zero and some transformation groups of complex spaces and Riemannian manifods are Jordan.

Cox构造的等变自同构及其应用

Equivariant automorphisms of the Cox construction and applications

本文刻画单扇Cox构造关于大环面中足够大的连通复李子群的等变全纯自同构群，并用于确定有理矩角流形及完备单扇环面簇的相应自同构群。