On the log–local principle for the toric boundary
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In this paper , 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.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. read more
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Open/closed correspondence via relative/local correspondence
TL;DR: In this article , Liu et al. established a correspondence between the disk invariants of a smooth toric Calabi-Yau 3-fold X with boundary condition specified by a framed Aganagic-Vafa outer brane (L,f) and the genus-zero closed Gromov-Witten invariants (GWCW,GWC) of a 4-fold toric X. This correspondence can be viewed as an instantiation of the log-local principle of van Garrel-Graber-Ruddat.
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Enumerative geometry of surfaces and topological strings
TL;DR: In this article , a survey of recent developments on the geometry and physics of Looijenga pairs is presented, including the log Gromov-Witten invariants of the pair.
References
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TL;DR: In this article, Gromov-Witten invariants of a smooth complex Deligne-Mumford stack with a projective coarse moduli space were introduced and proved.
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Toric degenerations of toric varieties and tropical curves
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