Spectral Data for SU(1,2) Higgs Bundles

Abstract

In this article we give an explicit description of the Hitchin fiber of SU(1,2) Higgs bundles $(L,F,\gamma,\beta)$ over a compact Riemann surface $X$ of genus $\ge 2$ with $q=\gamma\circ\beta$ having simple zeros and Toledo invariant $\tau=2 \rm{deg} L$ satisfying $|\rm{deg} L|<g-1$. In particular we identify the data in an SU(1,2) Higgs bundle as a Hecke transformation $\iota: F\to L^{-2}K\oplus LK$ at $Z(q)$. The Hitchin fiber is identified with a fiber bundle over $\text{Pic}^d X$ with unirational fiber, which is a GIT-quotient of a $\mathbb{C}^\times$-action on $\left(\mathbb{P}^1\right)^{4g-4}$. The base parametrizes choice of the line bundle $L$ and the fiber gives parameters for the Hecke transformation. The stable locus is shown to be a coarse moduli space of the corresponding moduli functor.