Relative Log-Symplectic structure on a semi-stable degeneration of moduli of Higgs bundles

TL;DR

This paper demonstrates that a semi-stable degeneration of Higgs bundle moduli spaces admits a relative log-symplectic form, extending Hitchin's symplectic structure to singular fibers and revealing integrable system structures.

Contribution

It introduces a relative log-symplectic form on the degeneration of Higgs moduli spaces, generalizing Hitchin's symplectic form to singular fibers with normal crossing singularities.

Findings

Existence of a relative log-symplectic form on the degeneration

Computation of Poisson ranks at all points

Description of the symplectic foliation on the closed fiber

Abstract

In a recent paper \cite{3}, a semi-stable degeneration of moduli space of Higgs bundles on a curve has been constructed. In this paper, we show that there is a relative log-symplectic form on this degeneration, whose restriction to the generic fibre is the classical symplectic form discovered by Hitchin. We compute the Poisson ranks at every point and describe the symplectic foliation on the closed fibre. We also show that the closed fibre, which is a variety with normal crossing singularities, acquires a structure of an algebraically completely integrable system.