Jeremy Daniel: Fundamental groups of smooth projective varieties and loop Hodge structures

Abstract:
It is an open problem to determine which finitely presented groups can be realized as the fundamental group of a smooth projective variety over the complex numbers. In the first part of the talk, I will explain some restrictions that one obtains from classical and non-abelian Hodge theory. In the second part, I will present the technique of variations of loop Hodge structures: these are infinite-dimensional vector bundles over a complex manifold with a lot of structures. This provides another point of view on non-abelian Hodge theory and can be used to give some applications to the topology and geometry of a (quasi-)projective variety.