A note on the cycle of curves in a product of pairs

TL;DR

This paper demonstrates that the cycle-valued logarithmic Gromov--Witten theory of a product of simple normal crossings pairs decomposes into a product of the theories of each factor, facilitating analysis via relative Gromov--Witten theory.

Contribution

It establishes a decomposition result for logarithmic Gromov--Witten invariants of product pairs using toroidal semistable reduction, linking it to relative Gromov--Witten theory.

Findings

Decomposition of cycle-valued logarithmic Gromov--Witten theory for products.

Application of toroidal semistable reduction to stabilization morphism.

Provides a pathway to analyze Gromov--Witten invariants via simpler components.

Abstract

We prove that the cycle-valued logarithmic Gromov--Witten theory of a product of simple normal crossings pairs $X\times Y$ decomposes into a product of pieces coming from $X$ and $Y$, provided that the decomposition is considered over a blowup of the moduli space of curves. The result is established by applying the toroidal semistable reduction theorem to the stabilization morphism on the stack of maps to the Artin fans. Since products of smooth pairs are naturally simple normal crossings pairs, the result provides a direct avenue of access to questions in logarithmic Gromov--Witten theory via relative Gromov--Witten theory.