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

Abstract

Abstract. 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 non-trivial cocycles α ∈ 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 coveringX 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.

Linear Shafarevich conjecture

“…Bogomolov and the second author suggest [BoKa98] that the Shafarevich conjecture might fail in the case of nonresidually finite fundamental groups. From another point of view, papers by Bogomolov and de Oliveira [BoDe0O5,BoDe0O6] suggest that big part of universal coverings of smooth projective varieties might still be holomorphically convex.…”