All Questions
Tagged with complex-geometry stein-manifolds
21 questions
11
votes
3
answers
1k
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 ...
- 1,450
10
votes
1
answer
808
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 $\...
- 3,383
9
votes
1
answer
935
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 ...
- 19.3k
8
votes
0
answers
284
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 ...
- 193
8
votes
0
answers
227
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.
...
- 181
7
votes
1
answer
291
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 ...
- 203
7
votes
0
answers
129
views
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|...
- 9,237
4
votes
1
answer
170
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
- 779
4
votes
1
answer
263
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 ...
- 28.9k
4
votes
0
answers
214
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 ...
- 1,170
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 ...
- 83
3
votes
2
answers
146
views
Vector bundles over a Stein space are projective
It is a "well known" fact that
locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules
(see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace ...
- 1,109
3
votes
0
answers
85
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 $...
- 615
2
votes
1
answer
687
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, ...
user21574
2
votes
1
answer
124
views
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 ...
- 779
2
votes
1
answer
709
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\...
- 14.3k
2
votes
0
answers
53
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)/...
- 615
2
votes
0
answers
109
views
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 ...
- 1,763
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 ...
- 779
2
votes
0
answers
196
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, ...
user113452
0
votes
0
answers
131
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\...
- 1,177