Strong Stein neighborhood bases

On the Steinness index

We introduce the concept of Steinness index related to the Stein neighborhood basis. We then show several results: (1) The existence of Steinness index is equivalent to that of strong Stein neighborhood basis. (2) On the Diederich-Fornæss worm domains in particular, we present an explicit formula relating the Steinness index to the well-known Diederich-Fornæss index. (3) The Steinness index is 1 if a smoothly bounded pseudoconvex domain admits finitely many boundary points of infinite type.

Good Stein Neighborhood Bases and Regularity of the d-bar Neumann Problem

We show that if the closure of a smooth pseudoconvex domain admits a "sufficiently nice" Stein neighborhood basis, then the d-bar Neumann problem on the domain is globally regular.

On regular Stein neighborhoods of a union of two maximal totally real subspaces in $\mathbb{C}^n$

We present a construction of regular Stein neighborhoods of a union of maximally totally real subspaces $M=(A+iI)\mathbb{R}^n$ and $N=\mathbb{R}^n$ in $\mathbb{C}^n$, provided that the entries of a real $n \times n$ matrix $A$ are sufficiently small. Our proof is based on a local construction of a suitable plurisubharmonic function $ρ$ near the origin, such that the sublevel sets of $ρ$ are strongly pseudoconvex and admit strong deformation retraction to $M\cup N$. We also give the application of this result to totally real immersions of real $n$-manifolds in $\mathbb{C}^n$ with only finitely …

Stein Domains in Complex Surfaces

Let S be a closed connected real surface and f a smooth embedding or immersion of S into a complex surface X. Assuming that the number of complex points of the immersion (counted with algebraic multiplicities) is non-positive we prove that f can be uniformly approximated by an isotopic immersion g whose image g(S) in X has a basis of open Stein neighborhoods which are homotopy equivalent to g(S). We obtain precise results for surfaces in the complex projective plane CP^2 and find an immersed symplectic sphere in CP^2 with a Stein neighborhood. Conversely, the generalized adjunction inequality …

Holomorphic fiber bundle with Stein base and Stein fibers

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

非仿射Stein流形与正规交叉除子

Non-Affine Stein Manifolds and Normal Crossing Divisors

该研究证明存在既非仿射也非拟射影却容许正规交叉除子紧化的Stein流形，借助接触几何类比的管道微积分研究相关邻域边界，并给出邻域由边界接触结构决定的条件。

On nonimbeddability of topologically trivial domains and Thin Hartogs figures of $P_2(\mathbb{C})$ into Stein spaces

A question of Poletsky was to know if there exists a thin Hartogs figure such that any of its neighborhoods cannot be imbedded in Stein spaces. In \cite{chirka}, Chirka and Ivashkovitch gave such an example arising in an open complex manifold. In this paper, we answer to the question of the existence of such a figure in compact surfaces by giving an example arising in $P_2(\mathbb{C})$. By smoothing it, we obtain a smooth (non analytic) disc with boundary $\bar{D} \subset P_2(\mathbb{C})$ having the same property. Consequently, this disc intersects all algebraic curves of $P_2(\mathbb{C})$. Mo…

通过拓扑同位素构造 Stein 曲面

Creating Stein surfaces by topological isotopy

本文结合 Freedman 拓扑与 Eliashberg 全纯理论，证明复曲面中驯服拓扑嵌入的 2-复形经小拓扑同位素后可作为不可数多个 Stein 正则邻域的交集，并借此研究拓扑伪凸性与伪凹性。

Stein domains and branched shadows of 4-manifolds

We provide sufficient conditions assuring that a suitably decorated 2-polyhedron can be thickened to a compact 4-dimensional Stein domain. We also study a class of flat polyhedra in 4-manifolds and find conditions assuring that they admit Stein, compact neighborhoods. We base our calculations on Turaev's shadows suitably "smoothed"; the conditions we find are purely algebraic and combinatorial. Applying our results, we provide examples of hyperbolic 3-manifolds admitting "many" positive and negative Stein fillable contact structures, and prove a 4-dimensional analogue of Oertel's result on inc…

An Example on $s$-H-Convexity in $\mathbb{C}^2$

We construct a bounded domain $Ω$ in $\mathbb{C}^2$ with boundary of class $\mathcal{C}^{1,1}$, such that $\overlineΩ$ has a Stein neighborhood basis, but is not $s$-H-convex for any real number $s\geq{1}$.

蠕虫域的自同构

Automorphisms of the Worm Domain

本文针对多复变中作为关键反例的Diederich-Fornæss蠕虫域开展研究，考察其自同构群，明确了该域边界除例外轨迹与端帽外处处局部球形的性质。

A note on plurisubharmonic defining functions in C^2

Let D be a smoothly bounded domain in C^2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis.

Exposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite 1-type

We show that for any bounded domain $Ω\subset\Cp ^n$ of 1-type $2k $ which is locally convexifiable at $p\in bΩ$, having a Stein neighborhood basis, there is a biholomorphic map $f:\barΩ\rightarrow \Cp ^n $ such that $f(p)$ is a global extreme point of type $2k$ for $f{(\barΩ)}$.

A note on plurisubharmonic defining functions in $\mathbb{C}^n$

Let D be a smoothly bounded domain in complex space of dimension larger than 2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis.

Proper holomorphic curves attached to domains

Let D be a domain in C^n with smooth boundary, of finite 1-type at a point p in the boundary and such that the closure of D has a basis of Stein Runge neighborhoods. Assume that there exists an analytic disc which intersects the closure of D exactly at p. We construct proper holomorphic maps from any open Riemann surface S which are attached to the closure of D exactly at p.

每个格罗莫夫椭圆域都是亚纯伪凸的

Every Gromov elliptic domain is meromorphically pseudoconvex

该研究证明每个格罗莫夫椭圆域都是亚纯伪凸的，明确了Stein流形中域的格罗莫夫椭圆性与Oka、Stein等性质的等价关系，还得出两类Oka流形非椭圆及射影空间中特定域非椭圆的结论。

Manifolds of holomorphic mappings from strongly pseudoconvex domains

Let D be a bounded strongly pseudoconvex domain in a Stein manifold S and let Y be a complex manifold. We prove that the graph of any continuous map from the closure of D to Y which is holomorphic in D admits a basis of open Stein neighborhoods in S x Y. Using this we show that many classical spaces of maps from the closure of D to Y which are holomorphic in D are infinite dimensional complex manifolds which are modeled on locally convex topological vector spaces (Banach, Hilbert or Frechet).

Extension of holomorphic bundles to the disc (and Serre's Problem on Stein bundles)

We show how to extend some holomorphic bundles with fifer C^2 and base an open set in C, to bundles on the Riemann Sphere, by an extremely simple technique. In particular, it applies to examples of non-Stein bundles constructed by Skoda and Demailly. It gives an example on C, with polynomial transition automorphisms.

Exposing boundary points of strongly pseudoconvex subvarieties in complex spaces

We prove that all locally exposable points in a Stein compact in a complex space can be exposed along a given curve to a given real hypersurface. Moreover, the exposing map for a boundary point can be sufficiently close to the identity map outside any fixed neighborhood of the point. We also prove a parametric version of this result for bounded strongly pseudoconvex domains in $\mathbb C^n$. For a bounded strongly pseudoconvex domain in $\mathbb C^n$ and a given boundary point of it, we prove that there is a global coordinate change on the closure of the domain which is arbitrarily close to th…

Cartan Theorems for Stein manifolds over a discrete valuation base

In this paper we prove Cartan theorems A and B for Stein manifolds over certain discrete valuation rings.