Higher genus relative and orbifold Gromov–Witten invariants

Tseng, Hsian-Hua; You, Fenglong

doi:10.2140/gt.2020.24.2749

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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 $(X, D)$ 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 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.

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Keywords

relative Gromov–Witten invariants, root stacks, degeneration, virtual localization

Mathematical Subject Classification 2010

Primary: 14N35

Secondary: 14H10

References

Publication

Received: 13 September 2018

Revised: 4 December 2019

Accepted: 2 January 2020

Published: 29 December 2020

Proposed: Dan Abramovich

Seconded: Richard P Thomas, Jim Bryan

Authors

Hsian-Hua Tseng
Department of Mathematics Ohio State University Columbus, OH United States

Fenglong You
Department of Mathematical and Statistical Sciences University of Alberta Edmonton, AB Canada

Volume 24, issue 6 (2020)