Higher genus relative and orbifold Gromov–Witten invariants

Abstract

Given a smooth projective variety $X$ and a smooth divisor $D\subset X$, we study relative Gromov–Witten invariants of $\left(X,D\right)$ and the corresponding orbifold Gromov–Witten invariants of the $r^{\text{ th}}$ root stack $X_{D,r}$. For sufficiently large $r$, we prove that orbifold Gromov–Witten invariants of $X_{D,r}$ are polynomials in $r$. Moreover, higher-genus relative Gromov–Witten invariants of $\left(X,D\right)$ are exactly the constant terms of the corresponding higher-genus orbifold Gromov–Witten invariants of $X_{D,r}$. We also provide a new proof for the equality between genus-zero relative and orbifold Gromov–Witten invariants, originally proved by Abramovich, Cadman and Wise (2017). When $r$ is sufficiently large and $X=C$ is a curve, we prove that stationary relative invariants of $C$ are equal to the stationary orbifold invariants in all genera.