Some results about holomorphic vector bundles over general Hopf manifolds
type: Item
instance of: scholarly article
title: Some results about holomorphic vector bundles over general Hopf manifolds
zbMATH Open document ID: 1210.32010
published in: Science in China. Series A
publication date: 2010-07-24T00:00:00Z
review text: A general Hopf manifold of dimension \(n\geq2\) is a manifold \(M\) of the form \(M:=W/G\), where \(W={\mathbb C}^n\setminus\{0\}\) and \(G=\pi_1(M)\). The fundamental group \(G\) has an infinite cyclic subgroup \({\mathbb Z}\) contained in the centre of \(G\) such that \(G={\mathbb Z}.K\) with \(K\) finite. When \(K\) is trivial, the Hopf manifold \(M\) is called primary. The purpose of this note is to give a brief survey of some results extending the theory of holomorphic bundles over primary Hopf manifolds to the general case. The results are a version of the Douady sequence, an isomorphism and vanishing theorem for the cohomology of \(\Omega^p(E)\) and a theorem stating that, if \(n\geq3\), the pullback of a holomorphic vector bundle of rank \(r\) to \(W\) is trivial if and only if the bundle is filtrable by subbundles.
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 32L05
Mathematics Subject Classification ID: 32J18
zbMATH DE Number: 5760414
zbMATH Keywords: holomorphic vector bundles
zbMATH Keywords: Hopf manifolds
zbMATH Keywords: cohomology groups
author: Xiangyu Zhou
reviewed by: P. E. Newstead
MaRDI profile type: Publication
full work available at URL: s11425-009-0223-3
OpenAlex ID: W1985522132
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DOI: 10.1007/S11425-009-0223-3
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description: scientific article; zbMATH DE number 5760414