Convexity of coverings of projective varieties and vanishing theorems

F. Bogomolov, B. De Oliveira

Published 2004-05-04Version 1

This article is concerned with the convexity properties of universal covers of projective varieties. We study the relation between the convexity properties of the universal cover of X and the properties of the pullback map sending vector bundles on X to vector bundles on its universal cover. Our approach motivates a weakened version of the Shafarevich conjecture. We prove this conjecture for projective varieties X whose pullback map identifies a nontrivial extension of a negative vector bundle $V$ by the trivial line bundle with the trivial extension. We prove the following pivotal result: if a universal cover of a projective variety has no nonconstant holomorphic functions then the pullback map of vector bundles is almost an imbedding. Our methods also give a new proof of the vanishing of the first cohomology for negative vector bundles $V$ over a compact complex manifold $X$ whose rank is smaller than the dimension of X.