Higher genus relative and orbifold Gromov-Witten invariants

TL;DR

This paper explores the relationship between higher genus relative and orbifold Gromov-Witten invariants, showing polynomial behavior in the orbifold case and linking relative invariants to constant terms of orbifold invariants, with new proofs and results for curves.

Contribution

It establishes polynomiality of orbifold Gromov-Witten invariants in the root parameter and relates higher genus relative invariants to orbifold invariants, providing new proofs and extending known results.

Findings

Orbifold Gromov-Witten invariants are polynomials in r for large r.

Higher genus relative invariants are the constant terms of orbifold invariants.

Stationary invariants of curves are equal in all genera for large r.

Abstract

Given a smooth projective variety $X$ and a smooth divisor $D\subset X$. We study relative Gromov-Witten invariants of $(X,D)$ and the corresponding orbifold Gromov-Witten invariants of the $r$-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 $(X,D)$ 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-Wise \cite{ACW}. 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.