Construction of the Poincar\'e sheaf on the stack of rank two Higgs bundles of $\mathbf{P}^{1}$

TL;DR

This paper constructs a maximal Cohen-Macaulay Poincaré sheaf on the stack of semistable rank two Higgs bundles over P^1, providing a foundational geometric object for further studies.

Contribution

It introduces the construction of a Poincaré sheaf on the stack of semistable rank two Higgs bundles on P^1, which is flat and Cohen-Macaulay.

Findings

Constructed the Poincaré sheaf on the stack of Higgs bundles.

Proved the sheaf is maximal Cohen-Macaulay.

Established flatness over the moduli stack.

Abstract

Let $\mathcal{H}iggs_{ss}$ be the stack of semistable rank two Higgs bundles on $\mathbf{P}^{1}$ with value in $O(n)$ where $n\geq 2$. In this paper we will construct the Poincar\'e sheaf $\mathcal{P}$ on $\mathcal{H}iggs_{ss}\times_{H}\mathcal{H}iggs_{ss}$ which is maximal Cohen-Macaulay and flat over $\mathcal{H}iggs_{ss}$.