Geometric Invariant Theory and Nonabelian Hodge Correspondence

The nonabelian Hodge correspondence is a deep and beautiful theory that identifies three moduli spaces: the Betti moduli space \(\mathcal{M}_{\text{B}}\) of surface group representations, the de Rham moduli space \(\mathcal{M}_{\text{dR}}\) of flat connections and the Dolbeault moduli space \(\mathcal{M}_{\text{Dol}}\) of Higgs bundles, whose objects on the surface appear to arise from very different contexts. The moduli spaces themselves are constructed as GIT quotients based on the geometric invariant theory pioneered by Mumford. The correspondence between these moduli spaces is the cumulative work of Corlette, Donaldson, Hitchin and Simpson built on that of many mathematicians and the machinery involved to prove these results lies in the intersection of algebraic geometry, complex geometry, geometric analysis and gauge theory.
The goal of this reading seminar is two-fold:

• Understand the workings of geometric invariant theory and how they are applied in the construction of the moduli spaces involved in the nonabelian Hodge correspondence.
• Understand the geometric properties of objects in the moduli spaces and without going into too much nasty detail, sketch the correspondence between the moduli spaces for \(G = \mathrm{GL}(n, \mathbb{C})\).

This topic lies in the intersection of many aspects of mathematics and may yield something useful for everyone in spite of different background and interests. For those more geometrically minded who are working with the moduli spaces, we provide the algebro-geometric foundation of their construction, while for those more algebraically minded, these are excellent examples of GIT and testing ground for your understanding. We aim to divide the topics in a way that strikes a balance between algebra and geometry, thereby bringing forth interaction and collaboration.

Date and time info
Thursdays, 11:00-12:30, starting 23.11.23

Keywords
Geometric Invariant Theory, GIT, Moduli problems, Nonabelian Hodge correspondence, character variety, flat connections, Higgs bundles

Prerequisites
Basic knowledge in Algebraic Geometry and Differential Geometry is useful but not required.