Moduli stacks of Higgs bundles on stable curves
Oren Ben-Bassat, Sourav Das, Tony Pantev
TL;DR
This work develops a robust framework for moduli of Higgs bundles in degeneration. By leveraging Jun Li’s stack of bounded expanded degenerations, it constructs a flat degeneration of the derived moduli stack of Higgs bundles on smooth curves and endows it with a relative log-symplectic structure that extends Hitchin’s form to the degeneration family. The Hitchin map is shown to have complete fibers and be flat, with a well-behaved reduced nilpotent locus that is isotropic in an open subset, enabling controlled dimension counts. The authors extend the relative logarithmic Dolbeault moduli over the universal moduli stack of stable curves, articulate log-structures and relative log-cotangent complexes along with their symplectic structures, and provide an extensive classical (Gieseker) atlas and dimension analysis in the Appendix, thereby linking derived and classical perspectives. Overall, the paper advances a degeneration-compatible, log-geometric approach to Hitchin systems and moduli of Higgs bundles with potential impacts on compactifications and shifted symplectic geometry in families.
Abstract
In this article, we construct a flat degeneration of the derived moduli stack of Higgs bundles on smooth curves using the stack of expanded degenerations of Jun Li. We show that there is an intrinsic relative log-symplectic form on the degeneration and we compare it with the one constructed by the second author. We show that the Hitchin map of the degeneration we construct has complete fibers. Furthermore, we show that the Hitchin map is flat and that a suitable open subset of the smooth locus of the reduced nilpotent cone is Lagrangian. We also extend the construction of the moduli of Higgs bundles along with the relative log-symplectic form over the universal moduli stack of stable curves.