Vanishing theorems of negative vector bundles on projective varieties and the convexity of coverings

Fedor Bogomolov, Bruno De Oliveira

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

We give a new proof of the vanishing of H1(X, V) for negative vector bundles V on normal projective varieties X satisfying rank V < dim X. Our proof is geometric, it uses a topological characterization of the affine bundles associated with nontrivial cocycles α ∈ H1 (X, V) of negative vector bundles. Following the same circle of ideas, we use the analytic characteristics of affine bundles to obtain convexity properties of coverings of projective varieties. We suggest a weakened version of the Shafarevich conjecture: the universal covering X̄ of a projective manifold X is holomorphically convex modulo the pre-image ρ-1(Z) of a subvariety Z ⊂ X. We prove this conjecture for projective varieties X whose pullback map ρ* identifies a nontrivial extension of a negative vector bundle V by script O sign with the trivial extension.

Original languageEnglish (US)
Pages (from-to)207-222
Number of pages16
JournalJournal of Algebraic Geometry
Volume15
Issue number2
DOIs
StatePublished - Apr 2006

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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