Virtual cycles on projective completions and quantum Lefschetz formula

TL;DR

This paper develops methods to localize virtual cycles on derived schemes and their projective completions, and applies these techniques to compare Gromov-Witten invariants of quintic threefolds, advancing the understanding of quantum invariants.

Contribution

It introduces new virtual cycle constructions on projective completions and projectivizations, relating them to Jiang-Thomas cycles and GW invariants of quintic threefolds.

Findings

Virtual cycles on projective completion push down to Jiang-Thomas cycles.

Difference of GW invariants expressed via virtual cycles on projectivizations.

New formulas for quintic and twisted quintic GW invariants.

Abstract

For a compact quasi-smooth derived scheme M with (-1)-shifted cotangent bundle N, there are at least two ways to localise the virtual cycle of N to M via torus and cosection localisations, introduced by Jiang-Thomas. We produce virtual cycles on both the projective completion and projectivisation and show the ones on the former push down to Jiang-Thomas cycles and the one on the latter computes the difference. Using similar ideas we give an expression for the difference of the quintic and t-twisted quintic GW invariants of Guo-Janda-Ruan.