Quantum Lefschetz property for genus two stable quasimap invariants

Abstract

By the reduced component in a moduli space of stable quasimaps to n-dimensional projective space ${\mathbb{P}}^n$ 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$\ge 3$. Then we prove the virtual fundamental cycle of the moduli space of stable quasimaps to a complete intersectionXin ${\mathbb{P}}^n$ of genus 2, degree $\ge 3$ is explicitly expressed in terms of the fundamental cycle of the reduced component of ${\mathbb{P}}^n$ and virtual cycles of lower genus $<2$ moduli spaces of X.