Fundamental Groups of Algebraic Varieties and the Shafarevich Conjecture

The fundamental group π1(X) is an important invariant of a complex algebraic variety X.  Despite its topological nature, it is closely connected to the geometry of many algebraic structures on X.  In this talk I want to discuss two elementary questions involving the fundamental group:

(1) How many simply-connected algebraic subvarieties does X contain?

(2) How many global holomorphic functions does the universal cover X̃

have?

The second question is related to a famous conjecture of Shafarevich, which asserts that in the case X is a smooth projective variety, X̃

has enough global holomorphic functions to separate points up to compact ambiguity.  I will describe joint work with Brunebarbe and Tsimerman which gives a complete answer to questions (1) and (2) provided π1(X)

admits a faithful linear representation, using techniques coming from non-abelian Hodge theory.