Asian Journal of Mathematics

Volume 9 (2005)

Number 3

Holomorphic Functions and Vector Bundles on Coverings of Projective Varieties

Pages: 295 – 314

DOI: http://dx.doi.org/10.4310/AJM.2005.v9.n3.a1

Authors

Fedor Bogomolov

Bruno De Oliveira

Abstract

Let $X$ be a projective manifold, $\rho:\tilde X \to X$ its universal covering and $\rho^*: Vect (X) \to Vect(\tilde X)$ the pullback map for the isomorphism classes of vector bundles. This article establishes a connection between the properties of the pullback map $\rho^*$ and the properties of the function theory on $\tilde X$. We prove the following pivotal result: if a universal cover of a projective variety has no nonconstant holomorphic functions then the pullback map $\rho^*$ is almost an imbedding.

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