Alexander Isaev
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.
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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.
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.
We classify all homogeneous Kobayashi-hyperbolic manifolds of dimension $n \ge 2$ whose group of holomorphic automorphisms has dimension either $n^2 - 7$ or $n^2 - 8.$ This paper continues the work of A. Isaev, who classified all such manifolds with automorphism group dimension $n^2 - 6$ and greater.
Large Harnack inequalities, Kobayashi distances and holomorphic motions
证明了多调和函数、全纯函数和几乎全纯函数的Harnack不等式的一些推广和类比。将所得结果应用于殆复流形上全纯运动光滑性的证明。
In this paper, we generalize the Kobayashi pseudo-distance to complex manifolds which admit holomorphic bracket generating distributions. The generalization is based on Chow's theorem in sub-Riemannian geometry. Let G be a linear semisimple Lie group. For a complex $G$-homogeneous manifold M with a G-invariant holomorphic bracket generating distribution D, we prove that (M,D) is Kobayashi hyperbolic if and only if the universal covering of M is a canonical flag domain and the induced distribution is the superhorizontal distribution.
The Schwarz lemma as one of the most influential results in complex analysis and it has a great impact to the development of several research fields, such as geometric function theory, hyperbolic geometry, complex dynamical systems, and theory of quasi-conformal mappings. In this note we mainly consider various version of Schwarz lemma and its relatives related to holomorphic functions including several variables.
We prove the Kobayashi-Hitchin correspondence and the approximate Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if $X$ is a compact manifold and $g$ is a Gauduchon metric on $X$, a twisted holomorphic vector bundle on $X$ is $g-$polystable if and only if it is $g-$Hermite-Einstein, and if $X$ is a compact Kähler manifold and $g$ is a Kähler metric on $X$, then a twisted holomorphic vector bundle on $X$ is $g-$semistable if and only if it is approximate $g-$Hermite-Einstein.
These are the notes of a short course I gave in the school "Aspects métriques et dynamiques en analyse complete", Lille, May 2015. The aim of this notes is to describe how to use a geometric structure (namely, the Kobayashi distance) to explore and encode analytic properties of holomorphic functions and maps defined on complex manifolds. We shall first describe the main properties of the Kobayashi distance, and then we shall present applications to holomorphic dynamics in taut manifolds, strongly pseudo convex domains and convex domains, and to operator theory in Bergman spaces (Carleson measu…
In this paper we study the following "slice rigidity property": given two Kobayashi complete hyperbolic manifolds $M, N$ and a collection of complex geodesics $\mathcal F$ of $M$, when is it true that every holomorphic map $F:M\to N$ which maps isometrically every complex geodesic of $\mathcal F$ onto a complex geodesic of $N$ is a biholomorphism? Among other things, we prove that this is the case if $M, N$ are smooth bounded strictly (linearly) convex domains, every element of $\mathcal F$ contains a given point of $\overline{M}$ and $\mathcal F$ spans all of $M$. More general results are pro…
Let $M$ be a Kobayashi hyperbolic homogenous manifold. Let $\mathcal F$ be a holomorphic foliation on $M$ invariant under a transitive group $G$ of biholomorphisms. We prove that the leaves of $\mathcal F$ are the fibers of a holomorphic $G$-equivariant submersion $π\colon M \to N$ onto a $G$-homogeneous complex manifold $N$. We also show that if $\mathcal Q$ is an automorphism family of a hyperbolic convex (possibly unbounded) domain $D$ in $\mathbb C^n$, then the fixed point set of $\mathcal Q$ is either empty or a connected complex submanifold of $D$.
Landau's theorem for holomorphic curves in projective space and the Kobayashi metric on hyperplane complements
我们证明了Dufresnoy定理的一个有效版本：对于n维复射影空间中任意一组2n+1个处于一般位置的超平面，我们找到一个显式常数K，使得对于从单位圆盘到这些超平面的补的全纯映射f，f在原点处关于Fubuni-Study度量的导数由K上界.这个结果给出了罗伊登函数的一个明确的下界，即，超平面补上的Kobayashi度量与Fubini-Study度量之比。我们的估计是基于Eremenko和Sodin的势理论方法。
Hyperbolic $n$-dimensional manifolds with automorphism group of dimension $n^2$
我们得到了维数为$n\ge 2$的复Kobayashi-双曲流形的一个完全分类，其中全纯自同构群的维数等于$n^2$。
The universal Kobayashi-Hitchin correspondence on Hermitian manifolds
我们证明了任意紧Hermitian流形上的一个非常一般的Kobayashi-Hitchin对应。这种对应关系是指“泛全纯定向对”的模空间。复几何中的大多数经典模问题（例如，G.具有约化结构群的全纯丛、全纯对、全纯Higgs对、Witten三元组、任意模问题）是这个普遍分类问题的特殊情况。我们的Kobayashi-Hitchin对应关系复杂的几何概念“多稳定定向全纯对”的存在性减少解决广义厄米-爱因斯坦方程。证明基于Uhlenbeck-Yau连续性方法。我们还研究了在这个一般的非Kaehler框架下的模空间的度量性质。我们更详细地讨论了定向联络的模空间、Douady Quot空间和非交换单极的模空间。
On the Kobayashi isometries of a class of 2-dimensional Lempert manifolds
本文研究具有至少三个元素的有限泛集的二维Lempert流形，证明其Kobayashi距离等距映射为全纯或反全纯，且无需正则性假设即为局部C^{1,1}类。
Estimates of the Kobayashi metric on almost complex manifolds
本文在殆复流形$(M,J)$上建立了Kobayashi-Royden无穷小伪度量的一个下界估计，其中该流形允许有界严格多重次调和函数.我们应用这个结果研究了$M$中严格伪凸域上度量的边界行为，并给出了$(M,J)$中域完全双曲的一个充分条件.最后得到了$M$中附着于全实子流形上的$J$ -全纯圆盘到边界的正则性。
Holomorphic maps acting as Kobayashi isometries on a family of geodesics
本文研究全纯映射在Kobayashi距离测地线族上作为等距映射时何时成为双全纯映射，在完全双曲区域和光滑有界拟凸区域两种情形下给出刻画，并通过例子证明假设的最优性。
We construct an unbounded strictly pseudoconvex Kobayashi hyperbolic and complete domain in $\mathbb{C}^2$, which also possesses complete Bergman metric, but has no nonconstant bounded holomorphic functions.
Hyperbolic 2-dimensional manifolds with 3-dimensional automorphism group
在本文中，我们确定了所有的Kobayashi-双曲二维复流形的全纯自同构群的维数为3。本文的工作总结了作者最近关于双曲$n$维流形的分类的一系列论文，其中自同构群的维数至少为$n^2-1$，其中$n\ge 2$。
Hyperbolic manifolds of dimension $n$ with automorphism group of dimension $n^2-1$
我们考虑复Kobayashi-双曲流形的维数为$n\ge 2$的全纯自同构群的维数等于$n^2-1$。我们给出了$n\ge 3$的这类流形的一个完整分类，并讨论了$n=2$的几个例子。
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。
Carleman approximation by holomorphic automorphisms of $\mathbb C^n$
利用$\mathbb C^n$的全纯自同构逼近非紧全实流形上的光滑映射。
Holomorphic automorphisms of Danielewski surfaces II -- structure of the overshear group
我们应用Nevanlinna理论代数簇Danielewski表面和调查他们的全纯自同构群。我们的主要结果是：已知在全纯自同构群的单位连通区稠密的过剪切群是自由的合并积。
Free dense subgroups of holomorphic automorphisms
证明了复Euklidean空间的全纯剪切群和过剪切群中存在由2个元素生成的自由稠密子群，并将这一结果推广到具有密度性质的Stein流形的全纯自同构群，只要存在广义平移.在全纯自同构的拓扑空间上，与这种广义平移相关的共轭算子是超循环的。
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).
Weakly holomorphic homogeneous regular manifolds
本研究提出弱全纯齐次正则流形（wHHR）这一新复流形类别，证明其包含全纯齐次正则流形等多类流形，并验证wHHR Stein流形上Bergman与Kobayashi度量双Lipschitz等价。
Given a sequence of automorphisms of the polydisk, we show that the associated composition semigroup homomorphisms on the ball of bounded holomorphic functions on the polydisk admit a universal inner function if a certain condition on the automorphisms are satisfied.
We consider the problem of interpolating a holomorphic family of nondegenerate holomorphic jets on C^n for n > 1, parametrized by points in a Stein manifold, by a holomorphic family of automorphisms of C^n. We show that under a suitable topological condition it is possible to find a solution, thereby generalizing results of Forstneric and Varolin concerning the nonparametric jet interpolation by automorphisms.
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。
Rigidity of Kobayashi isometries of a class of 2-dimensional Lempert manifolds
本文研究具有有限Carathéodory通用集的二维复流形的Kobayashi等距，证明其Kobayashi等距必为（反）全纯映射。
The Hermitian Connection and the Jacobi Fields of a Complex Finsler Manifold
证明了复Finsler流形的所有不变函数都可以由Kobayashi对复Finsler结构的全纯向量丛引入的联络的挠率和曲率完全恢复.本文还导出了一般复Finsler流形的测地线方程和Jacobi场方程，它们用Kobayashi联络表示。
The basin of attraction of a uniformly attracting sequence of holomorphic automorphisms that agree to a certain order in the common fixed point, is biholomorphic to $\mathbb{C}^n$. We also give sufficient estimates how large this order has to be.
An interpolation theorem for holomorphic automorphisms of {\bf C}$^n$
我们构造了$\C^n$的自同构，它将某些离散序列映射到另一个序列上，在每个点上具有指定的有限喷射，从而解决了自同构的一般Mittag-Leffler插值问题。在某些情况下，这可以在紧集上近似给定的自同构的同时完成。
On the automorphism groups of hyperbolic manifolds
证明了不存在维数为$n\ne 3$的Kobayashi双曲复流形，其全纯自同构群的维数为$n^2+1$，并且如果一个3维连通双曲复流形有维数为10的自同构群，则它与Siegel空间全纯等价.这些结果补充了作者以前关于复空间中整环自同构群的可能维数的定理。本文还证明了我们早先关于用$\ge n^2+2$维自同构群刻画$n$维双曲复流形的结果。
Holomorphic Factorization of Vector Bundle Automorphisms
该研究在复几何领域证明了Stein空间上秩2向量丛的零伦特殊全纯自同构可分解为有限个幂幺自同构或指数映射的乘积，推进了全纯向量丛自同构的结构研究。
This survey article mainly addresses to graduate students and young researchers in complex geometry willing to enter the beautiful word of connections between curvature and Kobayashi hyperbolicity. It is a detailed account of a recent breakthrough by Wu and Yau, shortly after generalized by Tosatti and Yang (and others), which sits on the crossroad between complex differential geometry and Kobayashi hyperbolicity. More specifically, an old conjecture by Kobayashi, stated at the very beginning of the theory, predicts that a compact hyperbolic manifold should have ample canonical bundle. Now, on…
We overview a number of precise notions for a holomorphic automorphism group to be big together with their implications, in particular we give an exposition of the notions of flexibility and of density property. These studies have their origin in the famous result of Andersén and Lempert from 1992 proving that the overshears generate a dense subgroup in the holomorphic automorphism group of $\C^n, n\ge 2$. There are many applications to natural geometric questions in complex geometry, several of which we mention here. Also the Linearization Problem, well known since the 1950 s and considered b…
Totally geodesic discs in strongly convex domains
证明了强凸区域之间的Kobayashi等距是全纯或反全纯的。更精确地说，令$n_1, n_2$是正整数，令$Ω_i \subset \C^{n_i}, \ i=1,2$，是有界$C^3$强凸区域。如果$ϕ: (Ω_1, d^K_{Ω_1}) \rightarrow (Ω_2, d^K_{Ω_2})$是等距的，即对于所有$ζ,η\in Ω_1,$都有$ d^K_Ω_{n_2}(f(ζ),f(η)) = d^K_{n_1} (ζ,η)$，则$ϕ$是全纯的或反全纯的。
Non-hyperbolicity of holomorphic symplectic varieties
本文证明了满足有理SYZ猜想的第二贝蒂数至少为5的本原辛簇的非双曲性，并进一步在第二贝蒂数至少为7时建立了Kobayashi伪度量恒为零的结果，从而完成了Kamenova-Lu-Verbitsky的工作。
For bounded pseudoconvex domains with finite type we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different points of the boundary, then the automorphism group has finitely many components and is the almost direct product of a compact group and connected Lie group locally isomorphic to ${ \rm Aut}(\mathbb{B}_k)$. Further, the limit set is a smooth submanifold diffeomorphic to the sphere of dimension $2k-1$. As applications we prove a new finite jet determination theorem and a Tits alternative theorem. The geometry of th…
In this paper we study when the Kobayashi distance on a Kobayashi hyperbolic domain has certain visibility properties, with a focus on unbounded domains. "Visibility" in this context is reminiscent of visibility, seen in negatively curved Riemannian manifolds, in the sense of Eberlein-O'Neill. However, we do not assume that the domains studied are Cauchy-complete with respect to the Kobayashi distance, as this is hard to establish for domains in $\mathbb{C}^d$, $d \geq 2$. We study the various ways in which this property controls the boundary behavior of holomorphic maps. Among these results i…
Introducing the deformation theory of holomorphic Cartan geometries, we compute infinitesimal automorphisms and infinitesimal deformations. We also prove the existence of a semi-universal deformation of a holomorphic Cartan geometry.
Realizing semisimple Lie groups as holomorphic automorphism groups of bounded domains
本文证明了包括所有连通实半单群在内的一大类李群（含许多非线性李群）均可实现为仿射复空间有界域的全纯自同构群，推广了1990年关于连通线性李群的结论。
Approximation of biholomorphic maps between Runge domains by holomorphic automorphisms
本文研究了复仿射空间中Runge域之间biholomorphic映射的近似问题，证明其可由全纯自同构逼近，并推广了相关结果。
We show that the group of all holomorphic automorphisms of complex affine space $\mathbb C^n$, $n>1$, and several of its subgroups satisfy the parametric Oka property with approximation and with interpolation on discrete sets.
Automorphisms of locally conformally Kahler manifolds
流形M是局部共形Kahler（LCK）的，如果它允许一个Kahler覆盖，其单值性由全纯同素作用。对于一个紧连通群G通过全纯自同构作用在LCK流形上，一个平均过程给出了一个G-不变的LCK度量。设U（1）作用在LCK流形M上的全纯同构，且该作用到M的Kahler覆盖的提升不是等距的.我们发现，覆盖承认自守Kahler潜在的，因此可以嵌入到一个霍普夫流形。
Let $X$ be a Stein manifold of dimension $n\geq 2$ satisfying the volume density property with respect to an exact holomorphic volume form. For example, $X$ could be $\mathbb{C}^n$, any connected linear algebraic group that is not reductive, the Koras-Russell cubic, or a product $Y\times\mathbb{C}$, where $Y$ is any Stein manifold with the volume density property. We prove that chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of $X$. In particular, $X$ has a chaotic holomorphic automorphism. A proof for $X=\mathbb{C}^n$ may be found in work of Fornæss and Sib…
Infinite-dimensionality of the Automorphism Groups of Homogeneous Stein Manifolds
我们证明了维数大于1的Stein流形X的全纯自同构群是无限维的，只要X是复李群的全纯作用的齐性空间.
Coherent state representations of the holomorphic automorphism group of a quasi-symmetric Siegel domain of type III
本文研究了III型拟对称Siegel域全纯自同构群的泛相干态表示分类，揭示了其与Kähler代数结构理论的联系，并提出基于广义Iwasawa分解的方法。
We show that each proper holomorphic self map of a symmetric power of the unit ball is an automorphism naturally induced by an automorphism of the unit ball, provided the ball is of dimension at least two.
Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$.
Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for Hölderian almost complex structures
若$X$是一个几乎复流形，具有类$\CC^α$的几乎复结构$J$，对某个$α>0$，对$p$上的每一点$p\in X$和每一切向量$V$，存在一个通过$p$的$J$ -全纯圆盘芽，且切向量为这个给定的切向量.这个存在的结果可以追溯到Nijenhuis-Woolf。在这种情况下，所有的$J$全纯曲线都是$\CC^{1,α}$类。然后，正如复流形一样，可以定义切向量的罗伊登-小林伪范数。问题是这个伪范数是否是切丛上的上半连续函数。对于复杂的流形，这是至关重要的一点罗伊登的证明等价的两个标准定义的小林伪度量。Kruglikov为足够光滑的结构建立了罗伊登-小林伪范数的上半连续性。在[I-R]中，证明了$J$的$\CC^{1,α}$正则性是足够的。在这里，我们展示了以下内容：定理。在单位双圆盘$\D^2\subset \C^2$上存在类$\CC^{1\over 2}$的几乎复结构$J$，使得Royden-Kobayashi范数不是切丛上的上半连续函数.
We construct canonical intertwining semi-models with Kobayashi hyperbolic base space for holomorphic self-maps of complex manifolds which are univalent on some absorbing cocompact hyperbolic domain. In particular, in the unit ball we solve the Valiron equation for hyperbolic univalent self-maps and for hyperbolic semigroups.
Hyperbolic metrics, homogeneous holomorphic functionals and Zalcman's conjecture
我们表明，使用小林和Caratheodory度量特殊的全纯磁盘在普遍的Teichmuller空间，一类广泛的全纯泛函的空间上的单叶函数在磁盘上的Koebe函数或其根变换最大化，他们的极值被迫双曲功能。作为结果，这意味着著名的Zalcman和比伯巴赫猜想的证明。
The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbf{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbf{C}^n\times \mathbf{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbf{C}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock-Bargmann-Hartogs domains in terms of the polylogarithm functions and Kim-Ninh-Yamamori determined the automorphism group of the domain $D_{n,m}(μ)$. In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock-Bargmann-Hartogs …
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…
Let $M$ be a complex manifold which admits an exhaustion by open subsets $M_j$ each of which is biholomorphic to a fixed domain $Ω\subset \mathbb C^n$. The main question addressed here is to describe $M$ in terms of $Ω$. Building on work of Fornaess--Sibony, we study two cases namely, $M$ is Kobayashi hyperbolic and the other being the corank one case in which the Kobayashi metric degenerates along one direction. When $M$ is Kobayashi hyperbolic, its complete description is obtained when $Ω$ is one of the following domains -- (i) a smoothly bounded Levi corank one domain, (ii) a smoothly bound…
The pentablock is a Hartogs domain over the symmetrized bidisc. The domain is a bounded inhomogeneous pseudoconvex domain, and does not have a $\mathcal{C}^{1}$ boundary. Recently, Agler-Lykova-Young constructed a special subgroup of the group of holomorphic automorphisms of the pentablock, and Kosiński completely described the group of holomorphic automorphisms of the pentablock. The purpose of this paper is to prove that any proper holomorphic self-mapping of the pentablock must be an automorphism.
Consider a holomorphic automorphism acting hyperbolically on an invariant compact set. It has been conjectured that the arising stable manifolds are all biholomorphic to Euclidean space. Such a stable manifold is always equivalent to the basin of a uniformly attracting sequence of maps. The equivalence of such basins to Euclideans has been shown under various additional assumptions. Recently Majer and Abbondandolo achieved new results by non-autonomously conjugating to normal forms on larger and larger time intervals. We show here that their results can be improved by adapting these time inter…
Inegalite d'Ahlfors en dimension superieure
证明了复紧致Kobayashi双曲流形上全纯曲线的一个Ahlfors不等式.
We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works of D. Guan and the first author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kähler manifold $W_F$ which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction, and prove that the automorphism group of $Q$ satisfies the Jordan proper…
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