Over a base variety X, nonabelian Hodge theory provides a correspondence among representations of the fundamental group of X, flat bundles, and Higgs bundles over X. The corresponding three moduli spaces (called Betti, de Rham, and Dolbeault, respectively) are also related to each other. In this talk, firstly I will give a brief introduction to this theory as the background setting, then I will report some recent work on exploring the geometry of these moduli spaces, more precisely: (1)Stratification of the de Rham moduli space (some conjectures of Simpson); (2)Dynamical systems on the Dolbeault moduli space; (3)Generalization of Deligne’s twistor construction; (4)Geometry of the base manifold which parametrizes a family of Higgs bundles.

1.14.pdf