Singular Lagrangian in the Hitchin moduli space and conformal limits

Abstract:

The C*-action on the Deligne moduli space of lambda-connections results in a Bialynicki-Birula type stratification which, restricted to the Dolbeault and de Rham moduli space respectively, gives the Morse and partial oper stratification. A stratum on the Dolbeaut moduli is the upward flow through a C*-fixed point. While the upward flow and the partial oper stratum are not related by the non-abelian Hodge correspondence, they admit a biholomorphic parametrization by a common vector space when the C*-fixed point is a non-singular point. The passage from the upward flow to the partial oper stratum through this parametrization is given by taking the conformal limit. In this case, both spaces are complex Lagrangians. More details will be given in the first part of the talk.

In the second part, we will look at the case when the C*-fixed point is singular. An analogous affine parametrization by a complex vector space is given which, upon taking quotient by the isotropy group of the C*-fixed point (in the GIT sense), represents a sublocus of the upward flow. The nonsingular part of the sublocus is shown to be Lagrangian. Finally, when the C*-fixed point lives in the open stratum or when its isotropy group is abelian, we demonstrate that the conformal limits of stable Higgs bundles over the aforementioned affine parametrization exist. If time permits, I will explain the main ideas behind the proof of the main results and discuss further directions. This work is based on part of my doctoral thesis.

Host: Parker Evans