Higher genus relative and orbifold Gromov–Witten invariants

Tseng, Hsian-Hua¹ ; You, Fenglong²

¹ Department of Mathematics, Ohio State University, Columbus, OH, United States
² Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada

Geometry & topology, Tome 24 (2020) no. 6, pp. 2749-2779.

Résumé

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.

Détail

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

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