Members' Colloquium
Fundamental Groups of Algebraic Varieties and the Shafarevich Conjecture
The fundamental group $\pi_1(X)$ 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 $\tilde{X}$ have?
The second question is related to a famous conjecture of Shafarevich, which asserts that in the case X is a smooth projective variety, $\tilde{X}$ 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 $\pi_1(X)$ admits a faithful linear representation, using techniques coming from non-abelian Hodge theory.