Oscar Garcia-Prada, ICMAT, Spain: Geometry of Hodge bundles on Riemann surfaces
Given a compact Riemann surface X and a complex reductive group G, we consider the moduli space of G-Higgs bundles over X. There is a natural C*-action on this moduli space given by scaling the Higgs field. The fixed points of this action with non-zero Higgs field are called Hodge bundles since, through the non-abelian Hodge correspondence, they are in bijection with variations of Hodge structure. In this talk, I will discuss several basic aspects of the geometry of Hodge bundles, including the recently introduced Toledo invariant of a Hodge bundle and its Arakelov-Milnor bound, as well as the condition of being very stable and its relation to mirror symmetry and Langlands duality.