Logahoric Higgs Torsors for a Complex Reductive Group

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 ${\mathcal{G}}_{\mathbf{\theta }}$-Higgs torsors on a smooth algebraic curve $X$, where ${\mathcal{G}}_{\mathbf{\theta }}$ is a parahoric group scheme on $X$. In the case when the group $G$ is the general linear group ${\mathrm{GL}}_n$, we show that the stability condition of a parahoric torsor is equivalent to the stability of a parabolic bundle. A correspondence between semistable logahoric ${\mathcal{G}}_{\mathbf{\theta }}$-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.

arXiv comment33 pages