Torelli conjecture for deformation-equivalent log Calabi–Yau pairs

Let $(Y,D)$ and $(Y',D')$ be $n$-dimensional, deformation-equivalent log Calabi–Yau pairs, where a log Calabi–Yau pair is a smooth projective variety together with an anticanonical reduced simple normal crossings divisor. Set $U=Y\setminus D$ and $U'=Y'\setminus D'$. Suppose that parallel transport induces an isomorphism

$$\mu\colon H^n(U)\cong H^n(U')$$

of mixed Hodge structures. Torelli conjecture for log Calabi–Yau pairs. Then there is an isomorphism of pairs

$$f\colon (Y,D)\cong (Y',D').$$

This is the expected Torelli principle for quasi-projective log Calabi–Yau varieties, asserting that their middle cohomology together with its mixed Hodge structure determines the pair within a deformation class. The paper proves a generic Torelli theorem for a class of three-dimensional log Calabi–Yau pairs with maximal boundary, while the statement here is formulated in general dimension.