Holomorphic functions and vector bundles on coverings of projective varieties
Let \(X\) be a projective manifold, \(\rho :\tilde{X}\rightarrow X\) its universal covering and \({\rho}^*:\text{Vect}(X)\rightarrow \text{Vect}(\tilde{X})\) the pullback map for the isomorphism classes of vector bundles. It is still unknown whether the non-compact universal covers of projective varieties do have non-constant holomorphic functions. The Shafarevich conjecture claims that the universal cover \(\tilde{X}\) of a projective variety \(X\) is holomorphically convex (i.e. \(\tilde{X}\) has a lot of non-constant holomorphic functions). In this paper, the authors prove the following interesting result: if a universal covering of a projective manifold \(X\) has no non-constant holomorphic functions, then the pullback map \({\rho}^*\) is almost an imbedding. This result can potentially be used to show that the universal covering of a projective manifold has a non-constant holomorphic function.
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