# Questions tagged [stein-manifolds]

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24
questions

**8**

votes

**1**answer

205 views

### Cotangent bundles of surfaces as varieties

As far as I understand, it is easy to see (and find in the literature) that the affine variety
$$z_1^2+z_2^2+z_3^2=1$$
with the restriction of the standard $\omega_{std}$ of $\mathbb{C}^3$ is ...

**14**

votes

**2**answers

578 views

### Affine (or Stein) tubular neighbourhood theorem

Fix an embedding $X\subset Y$ of smooth complex affine varieties, or Stein manifolds.
I would guess that in general there is no analytic neighbourhood $X\subset U\subset Y$ with a holomorphic ...

**0**

votes

**0**answers

59 views

### example of noncompact kahler manifold which is not modification of Stein manifold

It is obvious that every Stein manifold is Kahler. But the gap between these two can be quite huge. If the Kahler manifold contains compact complex submanifolds of positive dimension, it cannot be ...

**7**

votes

**0**answers

218 views

### Triviality of holomorphic vector bundles over $\mathbb{C}$

Let $E\longrightarrow\mathbb{C}$ be a holomorphic vector bundle.
I found two proofs that $E$ is trivial: one follows from Oka-Grauer principle (Theorem 5.3.1 in F. Forstneric, Stein Manifolds and ...

**6**

votes

**1**answer

155 views

### The state of art of the singular Levi problem — and hyperkähler quotients

One of the versions of the classical Levi problem asks the following:
Let $X$ be a complex manifold. Is it true that $X$ is Stein iff
$X$ admits a smooth exhaustion strictly plurisubharmonic ...

**6**

votes

**0**answers

180 views

### $T^*M$ is a Stein manifold: A clarification on the integral complex structure involved and its relation with the canonical symplectic form

I'm interested in understanding better the properties of the integrable (almost) complex structure that Eliashberg constructs on the cotangent bundle of a closed manifold $M$, whose dimension is at ...

**5**

votes

**0**answers

138 views

### Dense Stein subset in complex manifold

Let $X$ be a smooth proper algebraic variety. Then $X$ has a dense affine open subset. In particular, any smooth proper algebraic variety has a dense Stein open subset as the complement of a divisor.
...

**3**

votes

**0**answers

89 views

### Contact 3-manifolds with hyperkahler Stein fillings?

Is there any classification result on (homeomorphism type) of contact 3-manifolds $\Sigma$ that have Stein filling $W$ that is
1. Hyperkahler (s.t. Stein structure is the Kahler part of it)
2. not ...

**2**

votes

**0**answers

31 views

### Smoothings over a real interval

I am asking if somebody knows if the following kind of objects are studied somewhere or if there is some kind of obvious obstruction for them to appear.
Let $(X,0) \subset \mathbb{C}^n$ be a germ of ...

**4**

votes

**0**answers

64 views

### Extension of holomorphic function on family of relatively compact strictly pseudoconvex domains

Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and ...

**2**

votes

**0**answers

122 views

### Stein subspaces of polydiscs and balls

Let $D$ be a either an open polydisc or an open ball in $\mathbf{C}^n$.
(1) Let $\mathcal{O}$ be the $\mathbf{C}$-algebra of holomorphic functions on $\mathbf{C}^n$, resp. $D$, and let $f_1,\ldots, ...

**6**

votes

**1**answer

358 views

### Homology 3-sphere with a unique Stein-fillable contact structure

Are there any known examples of oriented integer homology 3-spheres $Y$ (besides $S^3$) which have exactly one Stein-fillable contact structure up to isotopy? Failing that, what are the known examples ...

**10**

votes

**3**answers

770 views

### Trivialisation of vector bundles on Stein spaces

Does every vector bundle on a Stein space have a finite local trivialisation?
Definitions:
Stein space means either a complex analytic Stein space or a nonarchimedean Stein space in the sense of ...

**4**

votes

**0**answers

246 views

### Is there a coordinate free proof of the Morrey--Kohn--Hormander identity?

The Morrey--Kohn--Hormander identity is the key to proving vanishing/existence results on bounded pseudoconvex domains in $\mathbb{C}^n$, or more generally, Stein domains. See, for instance, the ...

**1**

vote

**0**answers

74 views

### About interpolability of Stein structures

Imagine $V$ is a complex vector space and $U_1\subset U_2\subset V$ two Euclidean balls.
Let $\psi,\psi_1$ be strictly plurisubharmonic functions on $V$ and $U_1$ respectively.
Question: What are ...

**1**

vote

**1**answer

367 views

### what's the cohomological dimension of a Stein space?

I want to know the "cohomological dimension" of a Stein manifold.
I know that:
for $X$ differential manifold and for every sheaf $F$ of abelian
groups, I have $H_c^j(X,F)=H^j(X,F)=0$ for $j>\...

**2**

votes

**1**answer

614 views

### Cotangent bundle of coadjoint orbit is stein manifold?

Let me first define stein manifolds and coadjoint orbits.
A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold:
$X$ is holomorphically convex, ...

**9**

votes

**1**answer

544 views

### Vector bundles on Stein manifolds

This might be standard if true (if so, I shall be grateful if provided with a reference). Given a smooth map from a Stein manifold $X$ to $\operatorname{Gr}(k,n)$ (the Grassmannian of $k$ planes in $\...

**3**

votes

**1**answer

1k views

### Stein manifolds definiton

There are a few equivalent definitions of Stein manifolds. As far as I know they were initially defined as holomorphically convex complex manifolds, and then the other definitions (e.g. complex ...

**9**

votes

**1**answer

851 views

### Question about an estimate in Hörmander's proof of Cartan's Theorem B

I have been working through the proof of Cartan's Theorem B that Hörmander gives in his book 'Introduction to Complex Analysis in Several Variables'. When I began, I skipped over some of the initial ...

**2**

votes

**0**answers

405 views

### reference for p-adic Stein spaces

Hi,
I'm looking for a reference in english for p-adic Stein spaces. The usual referneces I come across are all in german.
Thanks

**2**

votes

**1**answer

541 views

### A basic question on the definition of Cartan-Remmert reduction and holomorphic convexity

Here is a definition of holomorphic convexity taken from the notes of Eyssidieux:
Defintion. A complex analytic space $S$ is holomorphically convex if there is a proper holomorphic morphism $\pi: S\...

**2**

votes

**1**answer

812 views

### Plurisubharmonic exhaustion functions without critical points at infinity

A complex manifold $X$ is said to be weakly pseudoconvex if there exists on $X$ a smooth plurisubharmonic exhaustion function $\psi$.
For example, Stein manifolds are weakly pseudoconvex (in this ...

**4**

votes

**1**answer

241 views

### Stein manifolds isomorphic at infinity

Suppose $M$ and $N$ are two Stein manifolds of dimension at least $3$ with compact subsets $U$ and $V$ such that $M\setminus U$ is biholomorphic to $N \setminus V$. It it true that $M$ is ...