Convexity of Coverings of Projective Varieties and Vanishing Theorems

Let X be a projective manifold, ρ : X̃ → X its universal covering and ρ∗ : V ect(X) → V ect(X̃) the pullback map for isomorphism classes of vector bundles. This article makes the connection between the properties of the pullback map ρ∗ and the properties of the function theory on X̃. Our approach motivates a weakened version of the Shafarevich conjecture: the universal covering X̃ of a projective manifold X is holomorphically convex modulo the pre-image ρ(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. We prove the following pivotal result: if a universal cover of a projective variety has no nonconstant holomorphic functions then the pullback map ρ∗ is almost an imbedding. Our methods also give a new proof of H(X,V ) = 0 for negative vector bundles V over a compact complex manifold X whose rank is smaller than the dimension of X.