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 $\alpha \in H^{1}(X,V)$ 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 $\tilde X$ of a projective manifold $X$ is holomorphically convex modulo the pre-image $\rho ^{-1}(Z)$ of a subvariety $Z \subset X$. We prove this conjecture for projective varieties $X$ whose pullback map $\rho ^{*}$ identifies a nontrivial extension of a negative vector bundle $V$ by $\mathcal {O}$ with the trivial extension.