Constructing Holomorphic Functions on Universal Coverings of Complex Algebraic Varieties

Which complex analytic spaces can arise as the universal cover of a complex algebraic variety? Motivated by this question, Shafarevich asked whether the universal cover of a smooth projective variety X is always holomorphically convex — that is, whether there exists a proper holomorphic map from the universal cover of X to a Stein space. This was established in the linear case — when the fundamental group of X admits an almost faithful complex linear representation — by Eyssidieux–Katzarkov–Pantev–Ramachandran, using tools from non-abelian Hodge theory. In this talk, I will discuss a generalization of Shafarevich’s question to the case of non-compact algebraic varieties. This is joint work with Ben Bakker and Jacob Tsimerman.