Higher genus relative and orbifold Gromov-Witten invariants

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.

arXiv comment29 pages. Revised according to referees' suggestions. Including results from arXiv:1804.09905 on target curves. To appear in Geometry & Topology

References in corpus (5)

• Orbifold Quantum Riemann-Roch, Lefschetz and Serre
• Double ramification cycles on the moduli spaces of curves
• Identification of the Givental formula with the spectral curve topological recursion procedure
• Gromov-Witten invariants of $\bp^1$ and Eynard-Orantin invariants
• Structures in genus-zero relative Gromov--Witten theory

Cited by in corpus (5)

• Double ramification cycles with target varieties
• Structures in genus-zero relative Gromov--Witten theory
• The local-orbifold correspondence for simple normal crossings pairs
• The local/logarithmic correspondence and the degeneration formula for quasimaps
• The log-open correspondence for two-component Looijenga pairs