Stability of Stein structures on the Euclidean space

Euclidean domains in complex manifolds

In this paper we find big Euclidean domains in complex manifolds. We consider open neighbourhoods of sets of the form $K\cup M$ in a complex manifold $X$, where $K$ is a compact $\mathscr O(U)$-convex set in an open Stein neighbourhood $U$ of $K$, $M$ is an embedded Stein submanifold of $X$, and $K\cap M$ is compact and $\mathscr O(M)$-convex. We prove a Docquier-Grauert type theorem concerning biholomorphic equivalence of neighbourhoods of such sets, and we give sufficient conditions for the existence of Stein neighbourhoods of $K\cup M$, biholomorphic to domains in $\mathbb C^n$ with $n=\dim…

Steiner-Minkowski Polynomials of Convex Sets in High Dimension, and Limit Entire Functions

For a convex set (K) of the (n)-dimensional Euclidean space, the Steiner-Minkowski polynomial (M_K(t)) is defined as the (n)-dimensional Euclidean volume of the neighborhood of the radius (t). Being defined for positive (t), the Steiner-Minkowski polynomials are considered for all complex (t). The renormalization procedure for Steiner polynomial is proposed. The renormalized Steiner-Minkowski polynomials corresponding to all possible solid convex sets in all dimensions form a normal family in the whole complex plane. For each of the four families of convex sets: the Euclidean balls, the cubes,…

Extending holomorphic mappings from subvarieties in Stein manifolds

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 …

Holomorphic submersions from Stein manifolds

In this paper we prove results on the existence and homotopy classification of holomorphic submersions from Stein manifolds to other complex manifolds. We say that a complex manifold Y satisfies Property S_n for some integer n bigger or equal the dimension of Y if every holomorphic submersion from a compact convex set in C^n of a certain special type to Y can be uniformly approximated by holomorphic submersions from C^n to Y. Assuming this condition we prove the following. A continuous map f from an n-dimensional Stein manifold X to Y is homotopic to a holomorphic submersions of X to Y if and …

带刺黎曼曲面的万有族是Stein流形

The universal family of punctured Riemann surfaces is Stein

本文证明了任意正数n下，亏格g的n-刺紧黎曼曲面的万有Teichmüller族是Stein流形，并研究了其函数论性质，包括纤维代数函数的稠密性、到欧氏空间的纤维代数嵌入以及相对Oka原理。

On holomorphic flows on Stein surfaces: transversality, dicriticalness and stability

We study the classification of the pairs $(N, \,X)$ where $N$ is a Stein surface and $X$ is a complete holomorphic vector field with isolated singularities on $N$. We describe the role of transverse sections in the classification of $X$ and give necessary and sufficient conditions on $X$ in order to have $N$ biholomorphic to $\mathbb C^2$. As a sample of our results, we prove that $N$ is biholomorphic to $\mathbb C^2$ if $H^2(N,\mathbb Z)=0$, $X$ has a finite number of singularities and exhibits a non-nilpotent singularity with three separatrices or, equivalently, a singularity with first jet …