A generic Torelli Theorem for certain log Calabi--Yau threefolds

Abstract

We prove a generic Torelli theorem for families $\pi :(\mathcal{Y},\mathcal{D})\to B$ of smooth three-dimensional log Calabi--Yau pairs, under the assumptions that one fiber $(Y,D)$ arises as the blowup of a toric pair, and $H^3(Y,\mathbb{Z})=0$. In the context of the Fano-LG correspondence, our result applies to very general fibers of families of Landau--Ginzburg models associated to a smooth Fano threefold with very ample anticanonical bundle.

Expositional improvements to Introduction and Section 2. Added section 4.4 with applications to mirror symmetry