Singular upward flows and conformal limits in the Hitchin moduli space

Abstract

The moduli space of Higgs bundles admits a $\mathbb{C}^{*}$-action, where taking the downward limits contracts the moduli space to the connected components of the fixed-point locus. Through a $\mathbb{C}^{*}$-fixed point, the locus which contracts to it is referred to as the upward flow through it. In this thesis, we construct a sublocus in the upward flow through a polystable fixed point called the central locus using the local Kuranishi model around the fixed point. We show that its intersection with the stable locus of the moduli space is a complex Lagrangian. When either the polystable $\mathbb{C}^{*}$-fixed point has a zero field or when its automorphism group is abelian, we prove that this intersection non-empty. Under the same assumption, we prove that the existence of the conformal limit of a stable point along the central locus. Finally, in rank two, we compute and give a concrete description of the central locus and the upward flow through a generic strictly polystable $\mathbb{C}^{*}$-fixed point. We compute the dimension of the upward flow and demonstrate that it has two components.