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arXiv · 2508.06750

Relative mirror symmetry, theta functions and the Gamma conjecture

Fenglong You
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Abstract

Let $X$ be a Fano variety, and $D\subset X$ be an snc anticanonical divisor. We study relative mirror symmetry for the log Calabi--Yau pair $(X,D)$. (1) We prove a relative mirror theorem for snc pairs without assuming the divisors are nef. (2) We study theta functions associated with the pair $(X,D)$. (3) We introduce functions on the mirror that are obtained from the higher-degree part of the big relative quantum cohomology. As an application, we use these new ingredients in relative mirror symmetry to prove a version of the mirror symmetric Gamma conjecture for $X$ for $\mathcal O_X$ and $\mathcal O_{\operatorname{pt}}$ in this setting, where the Landau--Ginzburg potential is defined as a sum of theta functions.

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Fenglong You. 2025-08-08. Relative mirror symmetry, theta functions and the Gamma conjecture. https://arxiv.org/abs/2508.06750

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