Complete integrability of the parahoric Hitchin system

TL;DR

This paper proves that the parahoric Hitchin system on a complex curve is completely integrable, showing the Hitchin map's fibers are abelian varieties and establishing key geometric properties of the moduli space.

Contribution

It establishes the complete integrability of the parahoric Hitchin system by analyzing the Hitchin map and the geometry of the moduli space of parahoric torsors.

Findings

Parahoric nilpotent cone is isotropic.

Moduli stack is 'very good' in the sense of Beilinson-Drinfeld.

Hitchin map is a Poisson map with abelian variety fibers.

Abstract

Let $\mathcal{G}$ be a parahoric group scheme over a complex projective curve $X$ of genus greater than one. Let $\mathrm{Bun}_{\mathcal{G}}$ denote the moduli stack of $\mathcal{G}$-torsors on $X$. We prove several results concerning the Hitchin map on $T^*\!\mathrm{Bun}_{\mathcal{G}}$. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that $\mathrm{Bun}_{\mathcal{G}}$ is "very good" in the sense of Beilinson-Drinfeld. We then prove that the parahoric Hitchin map is a Poisson map whose generic fibres are abelian varieties. Together, these results imply that the parahoric Hitchin map is a completely integrable system.