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 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.

AB - 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.

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U2 - 10.1090/S1056-3911-06-00428-0

DO - 10.1090/S1056-3911-06-00428-0

M3 - Article

AN - SCOPUS:33645376465

SN - 1056-3911

VL - 15

SP - 207

EP - 222

JO - Journal of Algebraic Geometry

JF - Journal of Algebraic Geometry

IS - 2

ER -