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 $\tau =2\mathrm{deg}\mathrm{L}$ satisfying $|\mathrm{deg}L|<\mathrm{g}-1$. 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.

References in corpus (2)

• Spectral data for G-Higgs bundles
• Cameral data for SU(p+1,p)-Higgs bundles