Let X be an irreducible reduced complex space on which a connected compact Lie group K acts by holomorphic automorphisms. Let G be the complexification of K and g the Lie algebra of G. Following the theory of algebraic transformation groups, we call the complex space X spherical if X is normal and its tangent space at some point is generated by the vector fields from a Borel subalgebra b or g. We give several characterizations of spherical Stein spaces. In particular, we prove that a connected Stein manifold X is spherical if and only if the algebra of K-invariant differential operators on X is commutative.
It is proved that a Stein manifold acted on by a connected compact Lie group is spherical if and only if there exists an antiholomorphic involution preserving each orbit of the action. This involution can be chosen equivariant with respect to a Weyl involution of the group.
In this new version, we give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in $\mathbb{C}^n, n\geq 2,$ is Kähler-Einstein if and only if the domain is biholomorphic to the ball. We establish versions of various classical theorems that are used in the solution for Stein spaces. Among other things, we construct a hyperbolic metric over a Stein space with spherical boundary. We also prove the Q. K. Lu type uniformization theorem for Stein spaces with isolated normal singularities.
The paper is a survey of recent results in geometric representation theory describing group actions which induce multiplicity-free representations in the spaces of holomorphic functions. For connected compact Lie groups of automorphisms of Stein manifolds we characterize such actions in terms of antiholomorphic involutions. Some proofs are given and some results are new. For example, spherical complex spaces are defined for arbitrary real forms of complex reductive groups. Their properties, which we prove here, were known only for compact real forms. We also show that a complex compact homogen…
We prove a complex version of Kostant's non-linear convexity theorem. Applications to the construction of G-invariant Grauert tubes of non-compact Riemannian symmetric G/K spaces are given.
Finite dimensional Stein spaces admitting a proper holomorphic embedding of the complex line are characterized, among all complex spaces, by their holomorphic endomorphism semigroup in the sense that any semigroup isomorphism induces either a biholomorphic or an antibiholomorphic map between them.
We discuss various known generalizations of the classical Hartogs' extension theorem on Stein spaces with arbitrary singularities and present an analytic proof based on d-bar methods.
We obtain new characterizations of Bergman and Bloch spaces on the unit disc involving equivalent (quasi)-norms of these spaces.Our results are in spirit of estimates obtained by Fefferman and Stein for HArdy spaces in R^n.
We prove that every reduced Stein space admits a holomorphic function without critical points. Furthermore, any closed discrete subset of such a space is the critical locus of a holomorphic function. We also show that for every complex analytic stratification with nonsingular strata on a reduced Stein space there exists a holomorphic function whose restriction to every stratum is noncritical. These result also provide some information on critical loci of holomorphic functions on desingularizations of Stein spaces. In particular, every 1-convex manifold admits a holomorphic function that is non…
Let X be a connected normal Stein space of pure dimension d>=2 with isolated singularities only. By solving a weighted d-bar-equation with compact support on a desingularization of X, we derive Hartogs' Extension Theorem on X by the d-bar-idea due to Ehrenpreis.
We give a characterization of the complex Euclidean space C^n, proving a global Newlander-Nirenberg's theorem on the representability of complex structures close to the standard complex structure on C^n.
The Cœuré-Loeb example as an open subset of a Stein space
University of Bucharest, Faculty of Mathematics and Computer Science（布加勒斯特大学数学与计算机科学学院）；Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西米翁·斯托伊洛瓦数学研究所）
In this article, we prove that if $Π: X\to Ω$ is a surjective holomorphic map, with $Ω$ a Stein space and $X$ a complex manifold of dimension $n\geq 3,$ and if, for every $x\in Ω$ there exists an open neighborhood $U$ such that $Π^{-1}(U)$ is Stein, then $X$ is Stein
In this paper we develop fundamental tools and methods to study meromorphic functions in an equivariant setup. As our main result we construct quotients of Rosenlicht-type for Stein spaces acted upon holomorphically by complex-reductive Lie groups and their algebraic subgroups. In particular, we show that in this setup invariant meromorphic functions separate orbits in general position. Applications to almost homogeneous spaces and principal orbit types are given. Furthermore, we use the main result to investigate the relation between holomorphic and meromorphic invariants for reductive group …
We show that the group of holomorphic automorphisms of a Stein manifold X of dimension greater than 1 is infinite-dimensional, provided X is a homogeneous space of a holomorphic action of a complex Lie group.
Suppose that Y is a complex manifold with the property that any holomorphic map from a compact convex set in a complex Euclidean space C^n (for any n) to Y is a uniform limit of entire maps from C^n to Y. We prove that a holomorphic map from a closed complex subvariety X_0 in a Stein manifold X to the manifold Y extends to a holomorphic map of X to Y provided that it extends to a continuous map. We then establish the equivalence of four Oka-type properties of a complex manifold. We also generalize a theorem of Siu and Demailly on the existence of open Stein neighborhoods of Stein subvarieties …
In section 1, we show that if $X$ is a Stein normal complex space of dimension n and $D\subset \subset X$ an open subset which is the union of an increasing sequence $D_{1}\subset D_{2}\subset ...\subset D_{n}\subset >...$ of domains of holomorphy in $X$. Then $D$ is a domain of holomorphy. In section 2, we prove that a domain of holomorphy $D$ which is relatively compact in a 2-dimensional normal Stein space $X$ itself is Stein. In section 3, we show that if $X$ is a Stein space of dimension n and $D\subset X$ an open subspace which is the union of an increasing sequence $D_{1}\subset D_{2}\s…
Let Z be a complex space and let S be a compact set in C^n x Z which is fibered over R^n (the real subspace of C^n). We give a necessary and sufficient condition for S to be a Stein compactum.
In this paper, we study the Bergman metric of a finite ball quotient $\mathbb{B}^n/Γ$, where $Γ\subseteq \mathrm{Aut}(\mathbb{B}^n)$ is a finite, fixed point free, abelian group. We prove that this metric is Kähler--Einstein if and only if $Γ$ is trivial, i.e., when the ball quotient $\mathbb{B}^n/Γ$ is the unit ball $\mathbb{B}^n$ itself. As a consequence, we establish a characterization of the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman-Einstein metric.
R. Guralnick [Linear Algebra Appl. 99, 85-96 (1988)] proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. In the preprints [arXiv:1703.09524] and [arXiv:1703.09530], a generalization of this to arbitrary (possibly, nonsmooth) 1-dimensional Stein spaces was obtained. The present paper contains a revised version of the proof from [arXiv:1703.09524]. The method of this revised proof can be used also in the higher dimensional case, which will be the subject of a forthcoming paper.
We give an abstract definition, similar to the axioms of a Stein manifold, of a class of complex Banach manifolds in such a way that a manifold belongs to the class if and only if it is biholomorphic to a closed split complex Banach submanifold of a separable Banach space.
We compute the Bass stable rank of the ring $Γ(X,\mathcal O_X)$ of global sections of the structure sheaf $\mathcal O_X$ on a finite-dimensional Stein space $(X,\mathcal O_X)$ and then apply this result to the problem of the factorization of invertible holomorphic matrices on $X$.
