On the log–local principle for the toric boundary

Abstract

Let X $X$ be a smooth projective complex variety and let D = D 1 + ⋯ + D l $D=D_1+\cdots +D_l$ be a reduced normal crossing divisor on X $X$ with each component D j $D_j$ smooth, irreducible and numerically effective. The log–local principle put forward in van Garrel et al. (Adv. Math. 350 (2019) 860–876) conjectures that the genus 0 log Gromov–Witten theory of maximal tangency of ( X , D ) $(X,D)$ is equivalent to the genus 0 local Gromov–Witten theory of X $X$ twisted by ⨁ j = 1 l O ( − D j ) $\bigoplus _{j=1}^{l}\mathcal {O}(-D_j)$ . We prove that an extension of the log–local principle holds for X $X$ a (not necessarily smooth) Q $\mathbb {Q}$ -factorial projective toric variety, D $D$ the toric boundary, and descendant point insertions.