Birational Invariance in Punctured Log Gromov-Witten Theory

TL;DR

This paper proves that punctured log Gromov-Witten invariants are invariant under certain log étale modifications, extending known birational invariance results and impacting mirror symmetry constructions.

Contribution

It generalizes birational invariance of Gromov-Witten theory to the punctured case and relates it to mirror algebra invariance under log étale modifications.

Findings

Punctured Gromov-Witten theory is invariant under log étale modifications.

Log étale invariance of logarithmic mirror algebras established.

Canonical scattering diagrams are preserved under these modifications.

Abstract

Given a log smooth scheme $(X,D)$, and a log \'etale modification $(\tilde{X},\tilde{D}) \rightarrow (X,D)$, we relate the punctured Gromov-Witten theory of $(\tilde{X},\tilde{D})$ to the punctured Gromov-Witten theory of $(X,D)$, generalizing results of Abramovich and Wise in the non-punctured setting in "Birational invariance in log Gromov-Witten Theory". Using the main comparison results, we show a form of log \'etale invariance for the logarithmic mirror algebras and canonical scattering diagrams constructed in "Intrinsic Mirror Symmetry" and "The Canonical Wall Structure and Intrinsic Mirror Symmetry" respectively.