On the log-local principle for the toric boundary

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 ${⨁}_{j=1}^l\mathcal{O}(-D_j)$. 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.

arXiv comment20 pages, slight generalisation of the main result

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• Sheaves of maximal intersection and multiplicities of stable log maps
• Log BPS numbers of log Calabi-Yau surfaces
• Symplectic cuts and open/closed strings I
• The local/logarithmic correspondence and the degeneration formula for quasimaps
• Enumerative geometry of surfaces and topological strings