Derived categories 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 the conjecture for a wide class of simple $K$-equivalences. This method relies on the construction of roofs of projective bundles introduced by Kanemitsu. Roofs are special Fano varieties of Picard number two admitting two projective bundle structures, and they are related to the construction of pairs of Calabi—Yau varieties: we prove that a $K$-equivalent pair is derived equivalent if the associated pair of Calabi—Yau varieties is derived equivalent, and we apply this technique on several cases. The proofs are based on the manipulation of semiorthogonal decompositions by mutations of exceptional objects.