On the Stein coverings for 1-convex spaces.
type: Item
instance of: scholarly article
title: On the Stein coverings for 1-convex spaces.
zbMATH Open document ID: 1033.32008
author: Tan Vovan
published in: Acta Mathematica Vietnamica
publication date: 2004-03-29T00:00:00Z
review text: This paper studies compactifications and modifications of complex varieties in the framework of \(1\)-convexity and Moishezon spaces. The theorems are as follows. Theorem 1.1. Let \(X\) be a \(1\)-convex space with exceptional set \(S\), \(L\to X\) a holomorphic line bundle. Then the following are equivalent. (a) \(L\) is weakly positive (i.e., a metric condition holds). (b) \(L\) is cohomologically positive (i.e., twists of any coherent analytic sheaf over \(X\) by high enough powers of \(L\) are acyclic). (c) \(L\) is ample. Theorem 2.1. Let \(X,S\) be as in Theorem\,1.1, and suppose that \(X\subset{\mathbb C}^k\times{\mathbb P}^\nu\) embeds in mixed linear space. Then \(X\) can be covered by \(q=\dim(S)+1\) of its Stein open subsets. The proof of Theorem 2.1 is by induction on \(\dim(S)\) and goes roughly as follows. The case \(\dim(S)=0\) is essentially tautological, and the claim of Theorem 2.1 holds if \(X\) is a hyperplane \(Y\) in \({\mathbb C}^k\times{\mathbb P}^\nu\), and in general one can look at a hyperplane section \(X\cap Y\) of \(X\) and thicken up a Stein open covering \(\{U_j\}\) in \(X\cap Y\) of \(X\cap Y\) by a theorem of Siu on Stein neighborhoods to an open Stein covering \(\{V_j\}\) of \(X\cap Y\) in \(X\), and add \(X\setminus(X\cap Y)\) to the covering, which also turns out to be Stein open in \(X\). Theorem 2.2. Let \(X\) be a complex algebraic space with \(H^1(X,F)\) finite dimensional for each coherent algebraic sheaf \(F\to X\). Let \(A\to X\) be a coherent algebraic sheaf, and \(A'\) the corresponding coherent analytic sheaf over the complex analytic space \(X'\) underlying \(X\). Then (a) \(X'\) is a \(1\)-convex space, and (b) the natural map of cohomology groups \(H^i(X,A)\to H^i(X',A')\) is bijective for \(i\geq1\). \S\,3 talks about affine covers of projectivizable \(1\)-convex algebraic spaces. The latter are complex spaces algebraically isomorphic to a space of the form \(M\setminus D\), where \(M\) is a projective algebraic spa…
reviewed by: Imre Patyi
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32F10
zbMATH DE Number: 1941864
zbMATH Keywords: Stein open coverings
zbMATH Keywords: 1-convex spaces
zbMATH Keywords: modifications
zbMATH Keywords: compactifications
zbMATH Keywords: Moishezon spaces
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1941864