What is nonabelian Hodge theory?

Abstract: Nonabelian Hodge theory can be thought as nonabelian analogue of (abelian) Hodge theory by replacing the abelian (coefficient) groups into nonabelian (coefficient) groups. This is mainly due to the celebrated work of Donaldson, Corlette, Hitchin, and Simpson, which gives us a correspondence between local systems and Higgs bundles. More precisely, the nonabelian Hodge theory gives an equivalence between the category of reductive representations of the fundamental group, the category of semisimple flat bundles, and the category of polystable Higgs bundles with vanishing rational Chern classes, through pluri-harmonic metrics. Moreover, such an equivalence of categories is functorial, and preserves tensor products, direct sums, and duals. In moduli viewpoint, this theory indicates that, the moduli space of irreducible representations (called character variety, or Betti moduli space), as a smooth affine variety, is complex analytic isomorphic to the moduli space of irreducible flat bundles (called de Rham moduli space), which is a smooth Stein manifold (in the sense of analytic topology), and is real analytic isomorphic to the moduli space of stable Higgs bundles (called Dolbeault moduli space), which is a smooth quasi-projective variety. All of these objects can be generalized to a family of flat λ-connections parametrized by λ ∈ C, a notion introduced by Deligne, further studied by Simpson, and Mochizuki.

In this talk, I will begin with a quick review of (abelian) Hodge theory as the motivation of this theory. Then I will introduce this theory precisely from an analytic viewpoint by introducing the work of Donaldson, Corlette, Hitchin, Simpson, and Mochizuki on the existence of pluri-harmonic metrics. Then I will talk about this theory from the moduli viewpoint. A good reference of this theory is a survey paper by S. Rayan and A. Garcı́a-Raboso ( “Introduction to nonabelian Hodge theory: flat connections, Higgs bundles, and complex variations of Hodge structure, Fields Inst. Monogr. 34 (2015), 131-171.”), you can also take the first chapter of my thesis as a reference.

algebraic geometrycombinatoricsdifferential geometrynumber theoryrepresentation theory

Audience: learners

What is ...? Seminar

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