Descendant log Gromov-Witten invariants for toric varieties and tropical curves

TL;DR

This paper establishes a correspondence between tropical curve counts and log Gromov-Witten invariants in toric varieties, extending to higher genus and explaining negative multiplicities in descendant invariants.

Contribution

It proves the genus zero and non-superabundant higher-genus correspondence between tropical and log Gromov-Witten invariants in toric varieties using degeneration techniques.

Findings

Confirmed the tropical-log Gromov-Witten correspondence for genus zero.

Extended the correspondence to certain higher-genus cases.

Clarified the role of negative multiplicities in descendant invariants.

Abstract

Using degeneration techniques, we prove the correspondence of tropical curve counts and log Gromov-Witten invariants with general incidence and psi-class conditions in toric varieties for genus zero curves and all non-superabundant higher-genus situations. We also relate the log invariants to the ordinary ones, in particular explaining the appearance of negative multiplicities in the descendant correspondence result of Mark Gross.