Higher genus reduced Gromov--Witten invariants via desingularizations of sheaves

TL;DR

The paper introduces two methods to desingularize sheaves on algebraic stacks, enabling the definition of reduced Gromov--Witten invariants across various GIT quotients and genera.

Contribution

It presents novel stack constructions that simplify sheaves, facilitating the computation of reduced Gromov--Witten invariants for broad classes of GIT quotients.

Findings

Two new stack constructions for sheaf desingularization.

Application of these constructions to define reduced Gromov--Witten invariants.

Framework applicable to all genera and a wide range of GIT quotients.

Abstract

Given $\mathfrak{F}$ a coherent sheaf on a Noetherian integral algebraic stack $\mathfrak{P}$, we give two constructions of stacks $\widetilde{\mathfrak{P}}$, equipped with birational morphisms $p:\widetilde{\mathfrak{P}}\to \mathfrak{P}$ such that $p^*\mathfrak{F}$ is simpler: in the Rossi construction, the torsion free part of $p^*\mathfrak{F}$ is locally free; in the Hu--Li diagonalization construction, $p^*\mathfrak{F}$ is a union of locally free sheaves. We use these constructions to define reduced Gromov--Witten invariants of a large class of GIT quotients in all genera.