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

Bogomolov, Fedor; de Oliveira, Bruno

Asian J. Math. 9, No. 3, 295-314 (2005).

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.

Reviewer: Vasile Brînzănescu (Bucureşti)

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

Keywords:

holomorphic functions; vector bundles; universal coverings; projective manifolds