Equivalences between Calabi–Yau manifolds and roofs of projective bundles

It is conjectured that many birational transformations, called K-inequalities, have a categorical counterpart in terms of an embedding of derived categories. In the special case of simple K-equivalence (or more generally K-equivalence), a derived equivalence is expected: we propose a method to prove derived equivalence for a wide class of such cases. This method is related to the construction of roofs of projective bundles introduced by Kanemitsu. Such roofs can be related to candidate pairs of derived equivalent, L-equivalent and non isomorphic Calabi–Yau varieties, we prove such claims in some examples of this construction. In the same framework, we show that a similar approach applies to prove derived equivalence of pairs of Calabi–Yau fibrations, we provide some working examples and we relate them to gauged linear sigma model phase transitions