## Abstract

We give a new proof of the vanishing of H^{1}(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 ∈ H^{1} (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 language | English (US) |
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Pages (from-to) | 207-222 |

Number of pages | 16 |

Journal | Journal of Algebraic Geometry |

Volume | 15 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2006 |

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology