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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

Given a smooth projective variety X and a smooth divisor D X, we study relative Gromov–Witten invariants of (X,D) and the corresponding orbifold Gromov–Witten invariants of the r th root stack XD,r. For sufficiently large r, we prove that orbifold Gromov–Witten invariants of XD,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 XD,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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relative Gromov–Witten invariants, root stacks, degeneration, virtual localization
Mathematical Subject Classification 2010
Primary: 14N35
Secondary: 14H10
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
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