Quantum Lefschetz property for genus two stable quasimap invariants

TL;DR

This paper proves the smoothness of the reduced component in genus 2 stable quasimap moduli spaces and expresses the virtual fundamental cycle of these spaces for complete intersections in terms of lower genus data.

Contribution

It establishes the smoothness of the reduced component in genus 2 and provides an explicit formula for the virtual cycle of stable quasimap spaces for complete intersections.

Findings

Reduced component is smooth in genus 2, degree ≥ 3.

Virtual fundamental cycle expressed in terms of lower genus cycles.

Explicit formula for quasimap invariants in genus 2.

Abstract

By the reduced component in a moduli space of stable quasimaps to n-dimensional projective space we mean the closure of the locus in which the domain curves are smooth. As in the moduli space of stable maps, we prove the reduced component is smooth in genus 2, degree greater or equal to 3. Then we prove the virtual fundamental cycle of the moduli space of stable quasimaps to a complete intersection X in the projective space of genus 2, degree greater or equal to 3 is explicitly expressed in terms of the fundamental cycle of the reduced component of the projective space and virtual cycles of lower genus moduli spaces of X.