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).
Richard Lärkäng's user avatar
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 ...
Mohan Ramachandran's user avatar
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 $...
Michael Albanese's user avatar
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 (...
Jonny Evans's user avatar
  • 6,935
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 ...
Mohan Ramachandran's user avatar
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 ...
Richard Thomas's user avatar
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 ...
Marco Golla's user avatar
  • 10.4k
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 ...
KhashF's user avatar
  • 2,588
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 ...
KSackel's user avatar
  • 1,101
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 ...
R W's user avatar
  • 16.6k
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....
Lev Soukhanov's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible