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

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.