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Mirror symmetry for log Calabi–Yau surfaces II

Part of: Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  22 October 2025

Jonathan Lai Open the ORCID record for Jonathan Lai [Opens in a new window]  and
Yan Zhou
Jonathan Lai
Affiliation:
jonathan.en.lai@gmail.com
Yan Zhou
Affiliation:
Department of Mathematics, Northeastern University, Boston, MA 02115, USA y.zhou@northeastern.edu
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Abstract

We show that the ring of regular functions of every smooth affine log Calabi–Yau surface with maximal boundary has a vector space basis parametrized by its set of integer tropical points and a $\mathbb {C}$-algebra structure with structure coefficients given by the geometric construction of Keel and Yu [The Frobenius structure theorem for affine log Calabi–Yau varieties containing a torus, Ann. Math. 198 (2023), 419–536]. To prove this result, we first give a canonical compactification of the mirror family associated with a pair $(Y,D)$ constructed by Gross, Hacking and Keel [Mirror symmetry for log Calabi–Yau surfaces I, Publ. Math. Inst. Hautes Ètudes Sci. 122 (2015), 65168] where $Y$ is a smooth projective rational surface, $D$ is an anti-canonical cycle of rational curves, and $Y\setminus D$ is the minimal resolution of an affine surface with, at worst, du Val singularities. Then, we compute periods for the compactified family using techniques from Ruddat and Siebert [Period integrals from wall structures via tropical cycles, canonical oordinates in mirror symmetry and analyticity of toric degenerations, Publ. Math. Inst. Hautes Ètudes Sci. 132 (2020), 1–82] and use this to give a modular interpretation of the compactified mirror family.

Keywords

algebraic geometrymirror symmetry

MSC classification

Primary: 14J33: Mirror symmetry
Secondary: 14J32: Calabi-Yau manifolds 14J10: Families, moduli, classification 14J26: Rational and ruled surfaces

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 8 , August 2025 , pp. 2025 - 2090
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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