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

The main result is an all-genus construction of reduced Gromov-Witten invariants for a large class of GIT quotients.This extends earlier results, limited to genus one and two.The main tool involves blowings-up of sheaves.More precisely, given F a coherent sheaf on a Noetherian integral algebraic stack P, we give two constructions of stacks z P, equipped with birational morphisms p W z P !P such that p F is simpler:(i) In the Rossi construction, the torsion-free part of p F is locally free.(ii) In the Hu-Li diagonalization construction, p F is a union of locally free sheaves.The construction in (i) above is an extension of the Nash blowup and certain flattenings of sheaves to Noetherian integral Artin stacks.We show that reduced Gromov-Witten invariants obtained from each of the constructions above coincide.1. Introduction 2599 2. Background 2608 3. Desingularizations of coherent sheaves 2612 4. Diagonalization 2620 5. Desingularization and diagonalization on stacks 2632 6. Components of abelian cones 2638 7. Application to stable maps 2646 8.