On Stein Neighborhood Basis of Real Surfaces

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…

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 …

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…

Stein紧集上代数簇的平展上同调

Étale cohomology of algebraic varieties over Stein compacta

本文证明Stein紧集上代数簇平展上同调与其解析化奇异上同调的比较定理，推导相关亚纯函数域上同调维数结论，并给出实解析空间紧子集上希尔伯特第17问题的定量版本。

Deformations of Stein structures and extensions of holomorphic mappings

Let X be a Stein manifold, A a closed complex subvariety of X, and f a continuous map from X to a complex manifold Y whose restriction to A is holomorphic. After a homotopic deformation of the Stein structure outside a neighborhood of A in X (and of its smooth structure when X is a Stein surface)we find a holomorphic map from X to Y which agrees with f on A and which is homotopic to f relative to A. The analogous results in the case when the variety A is empty have been obtained in the preprint math.CV/0507212.

A contractible Levi-flat hypersurface in C^2 which is a determining set for pluriharmonic functions

We construct a real analytic Levi-flat hypersurface M in a neighborhood of an ellipsoid B in C^2 such that the each leaf of the Levi foliation of M is a complex disc, M intersects the boundary of B transversely, and the intersection A of M with B has the following properties: (i) the closure of A is diffeomorphic to the ball in R^3 and has a basis of Stein neighborhoods in C^2, (ii) any real analytic function on A which is constant on each Levi leaf is constant, (iii) any pluriharmonic function in a connected open neighborhood of A and vanishing on A is identically zero.

Flat bundles over some compact complex manifolds

We construct examples of flat fiber bundles over the Hopf surface such that the total spaces have no pseudoconvex neighborhood basis, admit a complete Kähler metric, or are hyperconvex but have no nonconstant holomorphic functions. For any compact Riemannian surface of positive genus, we construct a flat $\mathbb P^1$ bundle over it and a Stein domain with real analytic bundary in it whose closure does not have pseudoconvex neighborhood basis. For a compact complex manifold with positive first Betti number, we construct a flat disc bundle over it such that the total space is hyperconvex but ad…

Constructing Stein manifolds after Eliashberg

A unified summary is given of the existence theory of Stein manifolds in all dimensions, based on published and pending literature. Eliashberg's characterization of manifolds admitting Stein structures requires an extra delicate hypothesis in complex dimension 2, which can be eliminated by passing to the topological setting and invoking Freedman theory. The situation is quite similar if one asks which open subsets of a fixed complex manifold can be made Stein by an isotopy. As an application of these theorems, one can construct uncountably many diffeomorphism types of exotic R^4's realized as …

The Oka principle for multivalued sections of ramified mappings

In this paper we establish a basic version of the Oka principle for multivalued sections of ramified holomorphic maps h from a complex manifold Z onto a Stein manifold X. If the ramification locus of h projects into a closed complex subvariety X' of X and if h admits a fiber dominating spray over a small neighborhood of any point in X\X' then any multivalued continuous section of h which is holomorphic in a neighborhood of X' and unramified in X\X' can be homotopically deformed to a global holomorphic multivalued section. The corresponding results for sections of unramified submersions were es…

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).

Holomorphic curves in complex spaces

We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex space admitting an exhaustion function whose Levi form has at least two positive eigenvalues at every point outside a compact set, and this condition is essential. The proof involves a lifting method for the boundary of the curve and a newly developed technique of gluing holomorphic sprays over Cartan pairs in Stein manifolds whose value lie in a complex space, with control up to the boundary of the domains…

On the algebraicization of certain Stein manifolds

To every real analytic Riemannian manifold M there is associated a complex structure on a neighborhood of the zero section in the real tangent bundle of M. This structure can be uniquely specified in several ways, and is referred to as a Grauert tube. We say that a Grauert tube is entire if the complex structure can be extended to the entire tangent bundle. We prove here that the complex manifold given by an entire Grauert tube is, in a canonical way, an affine algebraic variety. In the special case M = the 2-sphere, we show that any entire Grauert tube associated to a metric (not necessarily …

C^*- Actions on Stein analytic spaces with isolated singularities

Let $V$ be an irreducible complex analytic space of dimension two with normal singularities and $\vr:\mathbb{C^*}\times V\to V$ a holomorphic action of the group $\mathbb{C^*}$ on $V$. Denote by $\fa_\vr$ the foliation on $V$ induced by $\vr$. The leaves of this foliation are the one-dimensional orbits of $\vr$. %and its singularities are the fixed points of $\vr$. We will assume that there exists a \emph{dicritical} singularity $p\in V$ for the $\bc^*$-action, i.e. for some neighborhood $p\in W\subset V$ there are infinitely many leaves of $\mathcal {F}_\vr|_{W}$ accumulating only at $p$. The…

Complete hyperbolic neighborhoods in almost-complex surfaces

We prove that each point in an almost-complex surface has a basis of complete hyperbolic neighborhoods. The problem is local, and therefore we can consider the case when our surface is ${\bf R^4}$ with an arbitrary almost-complex structure $J$ of class $C^{1.α}$. Let $C$ be a non-singular $J$-complex curve passing through the origin. Our result cah be stated as follows: There exists a basis $\{U_j\}$ of neighborhoods of zero in ${\bf R^4}$, such that $(U_j,J)$ are complete hyperbolic in the sence of Kobayashi, moreover $(U_j\setminus C,J)$ are complete hyperbolic as well. The fact that this re…

On complex points of codimension 2 submanifolds

In this paper we study the structure of complex points of codimension 2 real submanifolds in complex $n$ dimensional manifolds. We show that the local structure of a complex point up to isotopy only depends on their type (either elliptic or hyperbolic). We also show that any such submanifold can be smoothly isotoped into a submanifold that has 2-strictly pseudoconvex neighborhood basis.

Modeling complex points up to isotopy

In this paper we examine the structure of complex points of real 4-manifolds embedded into complex 3-manifolds up to isotopy. We show that there are only two types of complex points up to isotopy and as a consequence, show that any such embedding can be deformed by isotopy to a manifold having 2-complete neighborhood basis.

The Complex Monge-Ampère Equation, Zoll Metrics and Algebraization

Let M be a real analytic Riemannian manifold. An adapted complex structure on $TM$ is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of $TM$. We prove here that the only real analytic Zoll metric on the $n$-sphere with an entire adapted complex structure on $TM$ is the round sphere. Using similar ideas, we answer a special case of an algebraization question raised by the first author, characterizing some Stein manifolds as affine algebraic in ter…