Holomorphic functions and vector bundles on coverings of projective varieties

Let X be a projective manifold, ρ: X → X its universal covering and ρ*: V ect(X) → V ect( X ) the pullback map for the isomorphism classes of vector bundles. This article establishes a connection between the properties of the pullback map ρ* and the properties of the function theory on X . 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.