#### Volume 24, issue 6 (2020)

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Higher genus relative and orbifold Gromov–Witten invariants

### Hsian-Hua Tseng and Fenglong You

Geometry & Topology 24 (2020) 2749–2779
DOI: 10.2140/gt.2020.24.2749
##### 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 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.

##### Keywords
relative Gromov–Witten invariants, root stacks, degeneration, virtual localization
Primary: 14N35
Secondary: 14H10