Holomorphic functions and vector bundles on coverings of projective varieties
type: Item
instance of: scholarly article
title: Holomorphic functions and vector bundles on coverings of projective varieties
zbMATH Open document ID: 1122.14033
author: Bruno De Oliveira
author: Fedor Bogomolov
published in: The Asian Journal of Mathematics
publication date: 2006-06-19T00:00:00Z
review text: Let \(X\) be a projective manifold, \(\rho :\tilde{X}\rightarrow X\) its universal covering and \({\rho}^*:\text{Vect}(X)\rightarrow \text{Vect}(\tilde{X})\) the pullback map for the isomorphism classes of vector bundles. It is still unknown whether the non-compact universal covers of projective varieties do have non-constant holomorphic functions. The Shafarevich conjecture claims that the universal cover \(\tilde{X}\) of a projective variety \(X\) is holomorphically convex (i.e. \(\tilde{X}\) has a lot of non-constant holomorphic functions). In this paper, the authors prove the following interesting result: if a universal covering of a projective manifold \(X\) has no non-constant holomorphic functions, then the pullback map \({\rho}^*\) is almost an imbedding. This result can potentially be used to show that the universal covering of a projective manifold has a non-constant holomorphic function.
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 32Q30
Mathematics Subject Classification ID: 32E05
zbMATH DE Number: 5033989
zbMATH Keywords: holomorphic functions
zbMATH Keywords: vector bundles
zbMATH Keywords: universal coverings
zbMATH Keywords: projective manifolds
reviewed by: Vasile Brînzănescu
MaRDI profile type: Publication
DOI: 10.4310/AJM.2005.V9.N3.A1
description: scientific article