# Questions tagged [stein-manifolds]

The stein-manifolds tag has no usage guidance.

38
questions

2
votes

0
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### Quasi-Stein spaces that are not Stein or affinoid

I'm looking for an example of a rigid analytic space, over a field containing the $p$-adic numbers which is complete with respect to a non-archimedean norm, that is quasi-Stein, in the sense of Kiehl, ...

3
votes

0
answers

97
views

### Stein manifolds admitting uniform strictly plurisubharmonic exhaustion functions

Let $(X,\omega)$ be a Kähler manifold. Call $X$ uniformly Stein with respect to $\omega$ if there exists an exhaustion map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\...

2
votes

0
answers

51
views

### Pullback of coherent sheaves on Stein manifolds

Let $i:X\to Y$ be a closed embedding of Stein spaces, $G$ be a coherent $O_Y$-module. Set $I=\ker(i^*:O(Y)\to O(X))$. Then $I$ is an ideal of the ring $O(Y)$. Is that true that $\Gamma(X,i^*G)=G(Y)/...

2
votes

0
answers

107
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### Looking for a proof of a result of Grauert and Kerner

I'm looking for a proof of the following result.
Let $X$ be a Stein manifold and $h: Z \to X$ be a holomorphic fibre bundle with a complex homogeneous fibre whose structure group is a complex Lie ...

3
votes

0
answers

82
views

### Intersection of Stein opens admits a Stein neighborhood basis?

Let $X$ be a Stein manifold, $K$ be a compact subset of $X$. Consider the following conditions:
1.$K$ admit an open neighborhood basis in $X$ whose members are Stein;
2.$K=\cap_{j\ge 1}V_j$, where $...

4
votes

0
answers

211
views

### Cartan–Remmert reduction of an algebraic variety

Let $V$ be a normal connected algebraic (say, quasi-projective) variety over complex numbers. Assume that underlying complex analytic space $V^\text{an}$ is holomorphically convex, and thus admits the ...

0
votes

0
answers

129
views

### Stein manifold homotopic to wedge of two Stein manifolds

I am not very conversant with Stein structure on a manifold so this may be a very silly question. Let $X$ and $Y$ be two Stein manifolds of dimension $n$, inside $\mathbb{C}^N$. Take $x\in X$ and $y\...

1
vote

0
answers

75
views

### Milnor fibration and Runge pair

Let $f:\mathbb{A}^3\to \mathbb{A}^1$ be a polynomial map. Let $0\in \mathbb{A}^1$ be critical value. If $c \in \mathbb{A}^1$ is very close to zero (c is a regular value), then for Milnor fibration we ...

3
votes

1
answer

181
views

### Stein fillable tight contact structures on the 3-torus

Kanda classified tight contact structures on the 3-torus. Which of them is Stein fillable? Is there any good reference?

2
votes

1
answer

122
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### Complex manifolds making Liouville fail

Let us consider $g\colon X\to Y$ holomorphic, where $X$ is a complex manifold and $Y$ is a Stein manifold.
I am searching for all the pairs $(X,Y)$ such that we can find some non constant $g$ with ...

7
votes

0
answers

126
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### holomorphic functions on Stein manifolds reaching maxima in a given point on the boundary (Shilov boundary)

Let $M$ be a Stein manifold with smooth, strictly
pseudoconvex boundary, and $x$ a point on its
boundary. Is there a holomorphic function $f$ on
$M$, smooth on the boundary, with strict
maximum of $|f|...

4
votes

1
answer

158
views

### Connectedness of boundary of a Stein domain

Let $Y$ be a Stein manifold and $D\subset\subset Y$ be a Stein domain. I think $\overline D$ has connected boundary, and it should be somewhere, but I cannot find a reference for this. Thanks

2
votes

0
answers

70
views

### Regular exposable points on the boundary of compacts in Stein manifolds

Given a Stein manifold $Y$, there exists $\rho$, a $\mathscr C^2$-smooth strictly plurisubharmonic exhausting function for $Y$, such that the set of critical points $C=\{z\in Y\;:\;d\rho(z)=0\}$ is ...

14
votes

1
answer

1k
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### Artin vanishing for Stein manifolds and restriction maps

In the setting of complex Stein manifolds $X$ of complex dimension $d$, the theorem of Andreotti--Frankel implies the vanishing of the singular cohomology group $H^i(X,\mathbb Z)=0$ for $i>d$. With ...

1
vote

1
answer

330
views

### How to compute singular homologies of affine hypersurface in $A^4$ [closed]

I was trying to compute singular homology in integer coefficient of the hypersurface $t^2-1=z^{n}+x(xy-1)$ contained in $A^4$. Can anyone help me computing that? Can anyone tell me some reference ...

3
votes

1
answer

314
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### How to get a Stein space which has homotopy type of suspension of another Stein space

Let $V^n$ be a Stein space(or Stein manifold) in $\mathbb{C}^N$. I want to construct a Stein space(or Stein manifold) $W^{n+1}$ such that $H_i(V;\mathbb{Z})=H_{i+1}(W; \mathbb{Z}).$
If we take the ...

7
votes

1
answer

286
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 ...

16
votes

2
answers

918
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 ...

8
votes

0
answers

278
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

257
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 ...

7
votes

0
answers

217
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 ...

8
votes

0
answers

221
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### 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

103
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### 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

34
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

70
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

190
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, ...

7
votes

1
answer

471
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 ...

11
votes

3
answers

1k
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### 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

283
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

76
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 ...

2
votes

1
answer

677
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, ...

10
votes

1
answer

786
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 $\...

4
votes

1
answer

2k
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### 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

923
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### 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

444
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### 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

696
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\...

3
votes

1
answer

984
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### 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

260
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 ...