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 -