Relative mirror symmetry for non-Fano varieties

Abstract

Given a smooth projective variety $X$ with a smooth anticanonical divisor $D$, we study mirror symmetry for the log Calabi--Yau pair $(X,D)$ without assuming that $D$ is nef. We consider the mirror proper Landau--Ginzburg model $(\check X,W)$ from the intrinsic mirror construction of Gross--Siebert. We examine the relationship between the regularized quantum period of $X$ and the classical period of $W$, and identify the discrepancy between them as originating from curve counts in $D$, governed by the mirror map associated with $D$. We also obtain an explicit formula for the proper potential $W$ that encodes this discrepancy. In the end, we show that the quantum period, together with the mirror map, gives exactly the same information as the proper potential.