An Analytic Construction of the Moduli Space of Higgs Bundles over Riemann Surfaces

Abstract

The moduli space of Higgs bundles over Riemann surfaces can be defined as a quotient of an infinite-dimensional space by an infinite-dimensional Lie group. In this thesis, we use the Kuranishi slice method to endow this quotient with the struc-ture of a normal complex space. We also give a direct proof that the moduli space is locally modeled on an affine geometric invariant theory quotient of a quadratic cone by a complex reductive group. Moreover, we show that the moduli space admits an orbit type decomposition such that the decomposition is a Whitney stratification, and each stratum has a complex symplectic structure and a Kähler structure. The complex symplectic structures glue to a complex Poisson bracket on the structure sheaf, and the Kähler structures glue to a singular weak Kähler metric on the moduli space. Finally, we use the symplectic cut to show that the moduli space admits a projective compactification and hence is quasi-projective.