Relative mirror symmetry, theta functions and the Gamma conjecture

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 ${\mathcal{O}}_{\mathrm{pt}}$ in this setting, where the Landau--Ginzburg potential is defined as a sum of theta functions.

73 pages. preliminary version. Comments are very welcome