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

Given $\mathfrak{F}$ a coherent sheaf on a Noetherian integral algebraic stack $\mathfrak{P}$, we give two constructions of stacks $\tilde{\mathfrak{P}}$, equipped with birational morphisms $p:\tilde{\mathfrak{P}}\to \mathfrak{P}$ such that $p^{\ast }\mathfrak{F}$ is simpler: in the Rossi construction, the torsion free part of $p^{\ast }\mathfrak{F}$ is locally free; in the Hu–Li diagonalization construction, $p^{\ast }\mathfrak{F}$ 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.