Absolute vs. Relative Gromov-Witten Invariants

TL;DR

This paper compares absolute and relative Gromov-Witten invariants in symplectic geometry, showing they coincide under certain conditions and providing examples of divergence, advancing understanding of invariants in positive genus cases.

Contribution

It establishes conditions under which absolute and relative Gromov-Witten invariants are equal, extending previous approaches to positive genus cases and identifying when they differ.

Findings

Invariants are equal when the hypersurface contains no relevant holomorphic curves.

They differ in a narrow range of dimensions and genera.

Examples demonstrate cases where invariants do not coincide.

Abstract

In light of recent attempts to extend the Cieliebak-Mohnke approach for constructing Gromov-Witten invariants to positive genera, we compare the absolute and relative Gromov-Witten invariants of compact symplectic manifolds when the symplectic hypersurface contains no relevant holomorphic curves. We show that these invariants are then the same, except in a narrow range of dimensions of the target and genera of the domains, and provide examples when they fail to be the same.