Vanishing Theorems of Negative Vector Bundles on Projective Varieties and the Convexity of Coverings

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 non-trivial 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 X̃ 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 O with the trivial extension.