A classical problem in complex algebraic geometry is understanding the topology of 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, Stover and Vidussi.