Profinite rigidity of Kähler groups

A classical problem in complex algebraic geometry is understanding the topology of closed complex submanifolds of complex projective space, so-called smooth complex projective varieties, and, more generally, of compact Kähler manifolds. Two natural topological invariants to consider are the fundamental group and its profinite completion; the latter is also known as the algebraic fundamental group. In this talk I will address the following questions: When is the fundamental group of a compact Kähler manifold uniquely determined by its profinite completion? And, when does the profinite completion even determine the homeomorphism type of the underlying manifold? In particular, I will explain positive answers to both questions in the case of a direct product of fundamental groups of closed hyperbolic Riemann surfaces. This talk is based on joint work with Hughes, Py, Spitler, Stover and Vidussi.