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 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.