Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



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

Authors: Fedor Bogomolov and Bruno de Oliveira
Journal: J. Algebraic Geom. 15 (2006), 207-222
Published electronically: January 11, 2006
MathSciNet review: 2199065
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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 $\alpha \in H^{1}(X,V)$ 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 $\tilde X$ of a projective manifold $X$ is holomorphically convex modulo the pre-image $\rho ^{-1}(Z)$ of a subvariety $Z \subset X$. We prove this conjecture for projective varieties $X$ whose pullback map $\rho ^{*}$ identifies a nontrivial extension of a negative vector bundle $V$ by $\mathcal {O}$ with the trivial extension.

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Fedor Bogomolov
Affiliation: Courant Institute for Mathematical Sciences, New York University, New York, New York 10012

Bruno de Oliveira
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124

Received by editor(s): December 1, 2003
Received by editor(s) in revised form: October 19, 2004
Published electronically: January 11, 2006
Additional Notes: The first author was partially supported by NSF grant DMS-0100837. The second author was partially supported by NSF Postdoctoral Research Fellowship DMS-9902393 and NSF grant DMS-0306487