Calabi-Yau fibrations, simple K-equivalence and mutations

A homogeneous roof is a rational homogeneous variety of Picard rank 2 and index $r$ equipped with two different ${\mathbb{P}}^{r-1}$-bundle structures. We consider bundles of homogeneous roofs over a smooth projective variety, formulating a relative version of the duality of Calabi–Yau pairs associated to roofs of projective bundles. We discuss how derived equivalence of such pairs can lift to Calabi–Yau fibrations, extending a result of Bridgeland and Maciocia to higher-dimensional cases. We formulate an approach to prove that the $DK$-conjecture holds for a class of simple $K$-equivalent maps arising from bundles of roofs. As an example, we propose a pair of eight-dimensional Calabi–Yau varieties fibered in dual Calabi–Yau threefolds, related by a GLSM phase transition, and we prove derived equivalence with the methods above.