### 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) |
---|---|

Pages (from-to) | 207-222 |

Number of pages | 16 |

Journal | Journal of Algebraic Geometry |

Volume | 15 |

Issue number | 2 |

State | Published - Apr 2006 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Journal of Algebraic Geometry*,

*15*(2), 207-222.

**Vanishing theorems of negative vector bundles on projective varieties and the convexity of coverings.** / Bogomolov, Fedor; De Oliveira, Bruno.

Research output: Contribution to journal › Article

*Journal of Algebraic Geometry*, vol. 15, no. 2, pp. 207-222.

}

TY - JOUR

T1 - Vanishing theorems of negative vector bundles on projective varieties and the convexity of coverings

AU - Bogomolov, Fedor

AU - De Oliveira, Bruno

PY - 2006/4

Y1 - 2006/4

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33645376465&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33645376465&partnerID=8YFLogxK

M3 - Article

VL - 15

SP - 207

EP - 222

JO - Journal of Algebraic Geometry

JF - Journal of Algebraic Geometry

SN - 1056-3911

IS - 2

ER -