Decomposition of degenerate Gromov-Witten invariants

TL;DR

This paper establishes a decomposition formula for logarithmic Gromov-Witten invariants in degenerating families, connecting algebraic geometry with tropical geometry through explicit formulas and examples.

Contribution

It introduces a virtual decomposition of the fiber of moduli stacks of stable logarithmic maps in degenerations, generalizing previous results to broader settings.

Findings

Decomposition formula for logarithmic Gromov-Witten invariants.

Explicit examples illustrating the formulas.

Generalization beyond normal crossings degenerations.

Abstract

We prove a decomposition formula of logarithmic Gromov-Witten invariants in a degeneration setting. A one-parameter log smooth family X->B with singular fibre over b_0 \in B yields a family M(X/B,\beta) -> B of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over b_0 in terms of rigid tropical curves. This generalizes one aspect of known results in the case that the fibre X_{b_0} is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.