# On the log-local principle for the toric boundary

## TL;DR

This paper extends the log-local principle to certain toric varieties, showing an equivalence between genus 0 log Gromov-Witten theory of maximal tangency and local Gromov-Witten theory with twists, including descendent insertions.

## Contribution

It generalizes the log-local principle to actorial projective toric varieties with toric boundary, incorporating descendent point insertions.

## Key findings

- Extended the log-local principle to actorial toric varieties.
- Proved the equivalence for genus 0 theories with maximal tangency.
- Included descendent point insertions in the equivalence.

## Abstract

Let $X$ be a smooth projective complex variety and let $D=D_1+\cdots+D_l$ be a reduced normal crossing divisor on $X$ with each component $D_j$ smooth, irreducible, and nef. The log-local principle of van Garrel-Graber-Ruddat conjectures that the genus 0 log Gromov-Witten theory of maximal tangency of $(X,D)$ is equivalent to the genus 0 local Gromov-Witten theory of $X$ twisted by $\bigoplus_{j=1}^l\mathcal{O}(-D_j)$. We prove that an extension of the log-local principle holds for $X$ a (not necessarily smooth) $\mathbb{Q}$-factorial projective toric variety, $D$ the toric boundary, and descendent point insertions.