## Holomorphic functions and vector bundles on coverings of projective varieties.(English)Zbl 1122.14033

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.

### MSC:

 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14E20 Coverings in algebraic geometry 32Q30 Uniformization of complex manifolds 32E05 Holomorphically convex complex spaces, reduction theory
Full Text: