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

Fedor Bogomolov, Bruno De Oliveira

Research output: Contribution to journalArticle

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

Fingerprint

Vanishing Theorems
Projective Variety
Vector Bundle
Convexity
Covering
Bundle
Trivial Extension
Pullback
Cocycle
Modulo
Circle

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

Cite this

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

In: Journal of Algebraic Geometry, Vol. 15, No. 2, 04.2006, p. 207-222.

Research output: Contribution to journalArticle

@article{711e2b410d984e688c50878057659237,
title = "Vanishing theorems of negative vector bundles on projective varieties and the convexity of coverings",
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.",
author = "Fedor Bogomolov and {De Oliveira}, Bruno",
year = "2006",
month = "4",
language = "English (US)",
volume = "15",
pages = "207--222",
journal = "Journal of Algebraic Geometry",
issn = "1056-3911",
publisher = "American Mathematical Society",
number = "2",

}

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 -