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arXiv · 2411.04067

Log Calabi-Yau mirror symmetry and non-archimedean disks

Sean Keel·Logan White·Tony Yue YU
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Abstract

We give a uniform construction that includes the mirror algebra to a smooth log Calabi-Yau variety with maximal boundary, proper over an affine variety, or to a one parameter maximal compact Calabi-Yau degeneration, as the spectrum of a commutative associative algebra with a canonical basis, whose structure constants are counts of non-archimedean analytic disks. More generally, we study the enumeration of non-archimedean analytic curves with boundaries, associated to a given transverse spine in the essential skeleton of the log Calabi-Yau variety. The moduli spaces of such curves are infinite dimensional. In order to obtain finite counts, we impose a boundary regularity condition so that the curves can be analytically continued into tori, that are unrelated to the given log Calabi-Yau variety. We prove the properness of the resulting moduli spaces, and show that the mirror algebra is a finitely generated commutative associative algebra, giving rise to a mirror family of log Calabi-Yau varieties.

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Sean Keel, Logan White, Tony Yue YU. 2024-11-06. Log Calabi-Yau mirror symmetry and non-archimedean disks. https://arxiv.org/abs/2411.04067

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