Bruno de Oliveira

Holomorphic functions on universal covers of projective manifolds.


Abstract: Problem 1: What are the strongest possible geometric properties of the algebra of global holomorphic functions on universal covers of projective manifolds?

Problem 2: Do universal covers of projective manifolds have nonconstant holomorphic functions?

We will approach these problems studying the fibers of the pullback map $\rho^*$ sending the moduli space of stable bundles on a projective manifold to the space of vector bundles on its universal cover. In relation to Problem 1, we generalize the Shafarevich conjecture and prove this generalization in certain cases. In respect to Problem 2 we show that the absence of holomorphic functions on the universal of a projective manifold would imply that the fibers of $\rho^*$ are discrete.