The moduli space of stable quotients

Abstract

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion of the quotient away from the singularities. New compactifications of classical spaces arise naturally: a nonsingular and irreducible compactification of the moduli of maps from genus 1 curves to projective space is obtained. Localization on the moduli of stable quotients leads to new relations in the tautological ring generalizing Brill-Noether constructions.The moduli space of stable quotients is proven to carry a canonical 2-term obstruction theory and thus a virtual class. The resulting system of descendent invariants is proven to equal the Gromov-Witten theory of the Grassmannian in all genera. Stable quotients can also be used to study Calabi-Yau geometries. The conifold is calculated to agree with stable maps. Several questions about the behavior of stable quotients for arbitrary targets are raised.

Orbifold quasimap theory

“…The theory of ε-stable quasimaps to a large class of GIT quotient targets was developed in [12], generalizing and unifying the earlier works [26,8,28,30]. In the appropriate general context of the theory, the GIT target is a smooth Deligne-Mumford stack (or orbifold), but to keep the technicalities at a reasonable level, [12] worked under the assumption that the GIT quotient is a smooth variety and delegated the orbifold case to subsequent work.…”

Virtual push-forwards

“…We show an analogue of the conservation of number for virtually smooth families. We show implications to Gromov-Witten invariants and give a new proof of a theorem of Marian, Oprea and Pandharipande [19] which compares the virtual classes of moduli spaces of stable maps and moduli spaces of stable quotients. …”

“…give a new proof to a theorem of Marian, Oprea, Pandharipande [19] which compares the virtual classes of moduli spaces of stable maps to projective spaces and moduli spaces of stable quotients.…”

Multi-point virtual structure constants and mirror computation of $CP^2$-model

“…The intersection number that appear at the right-hand side of (2.44) has been already computed in [1,15] and it is given as follows:…”