17
votes
Accepted
Affine (or Stein) tubular neighbourhood theorem
A theorem of Siu says there in fact exists such a retraction, see Corollary 1 in
Every Stein Subvariety Admits a Stein Neighborhood, Inventiones math. 38, 89-100 (1976).
13
votes
Accepted
Artin vanishing for Stein manifolds and restriction maps
The pairs (U,X) are called Runge pairs. The homology version of your statement is proved in the paper of
Andreotti and Narasimhan Annals of Math vol 76 no 3 (1962) 499-509 using Morse
Theory.The title ...
12
votes
Vector bundles on Stein manifolds
As has been established in the comments, the answer to your question is yes. It is a special case of a general result known as the Oka principle which has been strengthened over time. The key is that $...
8
votes
Accepted
Cotangent bundles of surfaces as varieties
This is not possible. There is something called the growth rate of symplectic cohomology which is subexponential for affine varieties and exponential for cotangent bundles of higher genus surfaces (...
7
votes
Accepted
Connectedness of boundary of a Stein domain
This is a consequence of the fact that any Stein manifold with complex dimension at least 2 has one end. This follows from Hartogs extension across compact sets in Stein manifolds. You can find a ...
6
votes
Affine (or Stein) tubular neighbourhood theorem
My original interest in this problem was the following question.
Given a holomorphic bundle $E$ on an affine/Stein scheme $X$ inside any scheme/analytic space $Y$ of finite embedding dimension, is it ...
6
votes
Accepted
How to get a Stein space which has homotopy type of suspension of another Stein space
I'm not sure what you mean by "suspension" of $V$ here. The notion of suspension I have in mind (doubling the cone of $V$ over its base) doesn't yield a manifold, and even if it did it would ...
5
votes
How to compute singular homologies of affine hypersurface in $A^4$
Denote the hypersurface $\{(x,y,z,t)\in\Bbb{C}^4\mid t^2-1=z^n+x(xy-1)\}$ by $X$. The equation $x=0$ defines a closed subset $Z$ of $X$ that can be identified with $C\times\Bbb{C}$ where $C$ is the ...
4
votes
Stein fillable tight contact structures on the 3-torus
Eliashberg proved only one of them is Stein fillable (and in fact it's the only one that's strongly fillable). The paper is quite short.
Y. Eliashberg, Unique holomorphically fillable contact ...
2
votes
Complex manifolds making Liouville fail
In what concerns covers of compact complex manifolds there are results of Lin and Lin - Zaidenberg. In particular, nilpotent (more generally, FC-hypernilpotent) covers are always Liouville, whereas ...
1
vote
The state of art of the singular Levi problem -- and hyperkähler quotients
Privet, Anya.
Your reference [FN80] actually seems to contain an answer to this problem! They state (in particular, see the question 1.5 in the introduction) that the class of weakly psh functions, i....
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