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

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 epsilon 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 of a projective manifold X is holornorphically convex modulo the pre-image p(-1)(Z) of a subvariety Z subset of X. We prove this conjecture for projective varieties X whose pullback map p* identifies a nontrivial extension of a negative vector bundle V by O with the trivial extension.