Local Gromov-Witten Invariants are Log Invariants

TL;DR

This paper establishes an equivalence between two types of virtual counts of rational curves in specific geometric settings, providing a new perspective on Gromov-Witten invariants and their relation to log invariants.

Contribution

It proves a simple equivalence between local Gromov-Witten invariants and log invariants for anti-nef line bundles, and conjectures a generalization to sums of line bundles.

Findings

Proved equivalence between local Gromov-Witten invariants and log invariants.

Established a relation for rational curves in the total space of anti-nef line bundles.

Conjectured a broader generalization to direct sums of line bundles.

Abstract

We prove a simple equivalence between the virtual count of rational curves in the total space of an anti-nef line bundle and the virtual count of rational curves maximally tangent to a smooth section of the dual line bundle. We conjecture a generalization to direct sums of line bundles.