Logahoric Higgs Torsors for a Complex Reductive Group

TL;DR

This paper introduces and studies the stability of logahoric Higgs torsors for complex reductive groups on algebraic curves, establishing a moduli space with a Poisson structure.

Contribution

It defines logahoric Higgs torsors, introduces a stability notion, and constructs their moduli space with a Poisson structure, extending known results for parabolic bundles.

Findings

Stability of logahoric Higgs torsors is equivalent to parabolic bundle stability for GL(n).

A correspondence between semistable torsors and equivariant Higgs bundles is established.

The moduli space of these torsors admits an algebraic Poisson structure.

Abstract

In this article, a logahoric Higgs torsor is defined as a parahoric torsor with a logarithmic Higgs field. For a connected complex reductive group $G$, we introduce a notion of stability for logahoric $\mathcal{G}_{\boldsymbol\theta}$-Higgs torsors on a smooth algebraic curve $X$, where $\mathcal{G}_{\boldsymbol\theta}$ is a parahoric group scheme on $X$. In the case when the group $G$ is the general linear group ${\rm GL}_n$, we show that the stability condition of a parahoric torsor is equivalent to the stability of a parabolic bundle. A correspondence between semistable logahoric $\mathcal{G}_{\boldsymbol\theta}$-Higgs torsors and semistable equivariant logarithmic $G$-Higgs bundles allows us to construct the moduli space explicitly. This moduli space is shown to be equipped with an algebraic Poisson structure.