2004/05/04 by Fedor Bogomolov, F. Bogomolov, Bruno De Oliveira +3
Mathematics · #14E20 #14F05 #32E05 #32Q30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #math.AG #math.CV #msc:14E20 #msc:14F05 #msc:32E05 #msc:32Q30
paper · pdf · doi:10.48550/arxiv.math/0405066
arxiv created 2004/05/04 · openalex publication_date 2004/05/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.