Construction of the Poincare sheaf for higher genus curves

TL;DR

This paper constructs a Poincare sheaf on the moduli stack of rank two Higgs bundles over higher genus curves, extending the understanding of spectral data and sheaf properties in this geometric setting.

Contribution

It introduces a construction of the Poincare sheaf on the Higgs moduli stack for higher genus curves, including nonreduced spectral curves, as a maximal Cohen-Macaulay sheaf.

Findings

Poincare sheaf constructed on Higgs moduli stack

Sheaf is maximal Cohen-Macaulay and flat over the semistable locus

Includes nonreduced spectral curves

Abstract

Let $X$ be a smooth projective curve of genus $g$ and $L$ be a degree $l$ line bundle on $X$ with $l\geq 2g-1$. Denote the stack of rank two Higgs bundles on $X$ with value in $L$ by $\mathcal{H}iggs$ and the semistable part by $\mathcal{H}iggs_{ss}$. Let $H$ be the Hitchin base. In this paper we will construct the Poincare sheaf $\mathcal{P}$ on $\mathcal{H}iggs\times_{H}\mathcal{H}iggs_{ss}$ which is a maximal Cohen-Macaulay sheaf and flat over $\mathcal{H}iggs_{ss}$. In particular this includes the locus of nonreduced spectral curves.