Abstract:

Abstract: We will start by looking at the basic setup of nonabelian Hodge theory using higher nonabelian cohomology stacks via simplicial presheaves on the site of schemes. The de Rham, Betti and Dolbeault cohomology stacks are defined and related. The de Rham to Dolbeault degeneration glues, via the Riemann-Hilbert correspondence, to create the twistor space. We then specialize to the character variety and look at how Hitchin's equations give rise to prefered sections and the hyperkahler structure. Parabolic structures for quasiprojective varieties fit into a weight yoga. The higher nonabelian cohomology stacks give structures relative to the character variety. Recent work includes an investigation of what happens near infinity in the character variety.