Limiting Configurations for the SU(1,2) Hitchin Equation

Abstract

This dissertation studies the SU(1,2) Higgs bundle and a limiting behavior of solutions of the SU(1,2) Hitchin's self-duality equation. On a closed Riemann surface $X$ of genus $g\ge 2$, an SU(1,2) Higgs bundle consists of the following data: a rank two holomorphic vector bundle $F$ and the holomorphic maps $\beta: L\otimes K_X^{-1}\to F$, $\gamma: F\to L\otimes K_X$ where $L=\det F^\ast$. The Hitchin map of the moduli space of SU(1,2) Higgs bundles takes $(F,\beta,\gamma)$ to the quadratic differential $q=\gamma\circ\beta$. For an SU(1,2) Higgs bundle $(F,\beta,\gamma)$, the Hitchin equation is a non-linear PDE of hermitian metric $h$ on $F$. The existence of a unique solution follows from the stability condition. For a stable SU(1,2) Higgs bundle $(F,\beta,\gamma)$, we give an explicit description of the behavior of $h_t$, the unique solution of SU(1,2) Hitchin equation for the family $(F,t\beta,t\gamma)$ in the case where $q$ has simple zeros in the limit $t\to\infty$. In Chapter 1, we review the notion of $G$ Higgs bundles and focus on the case $G=$SU(1,2). The simple zeros of $q=\gamma\circ\beta$ are one of the three types: (1) a zero of $\beta$, (2) a zero of $\gamma$, or (3) neither. We present a stability condition in terms of the number of zeros of each type. We also review notions of the filtered bundle and the wild harmonic bundle. In Chapter 2, we give an explicit description of the fiber of the Hitchin map in terms of a fiber bundle over the Jacobian of $X$ with unirational fibers. The fiber is a GIT quotient of a $\mathbb{C}^\times$-action on $(\mathbb{P}^1)^{4g-4}$. The base parametrizes the choice of a line bundle $L$. The fiber gives parameters for a Hecke modification $\iota: F\to V$ which realizes $F$ as a rank-two locally free subsheaf of $V=L^{-2}K_X\oplus LK_X$. We show that the stable locus is a coarse moduli space of the appropriate moduli functor. In Chapters 3 and 4, we study the Hitchin equation for the family $(F,t\beta,t\gamma)$ as $t\to\infty$. In particular, we show that the limiting configuration $h_\infty$ satisfies the decoupled Hitchin equation and is induced from a harmonic metric $h_L$ on $L$ via the Hecke modification $\iota: F\to V$. The metric $h_L$ is adapted to a filtered line bundle $(L,\underline{\lambda_\infty})$ where the weights $\underline{\lambda_\infty}$ are specified by a rule depending on the types of zeros and their count. We prove the convergence of $h_t$ to $h_\infty$ after appropriate normalizing by gluing local model solutions constructed from wild harmonic bundles on $\mathbb{P}^1$ over disks around the zeros to a solution of the decoupled equation on the complement.