Derived equivalence for the simple flop of type $D_5$

We prove that every simple flop of type $D_5$, i.e., resolved by blowups with exceptional divisor isomorphic to a generalized Grassmann bundle with fiber $OG(4,10)$, induces a derived equivalence. This provides new evidence for the DK conjecture of Bondal–Orlov and Kawamata. The proof is based on a sequence of mutations of exceptional objects: we use the same argument to prove derived equivalence for some pairs of non-birational Calabi–Yau fivefolds in $OG(5,10)$, related to Manivel’s double–spinor Calabi–Yau varieties. We extend the construction to prove the derived equivalence of Calabi–Yau fibrations, which are described as zero loci in some generalized Grassmann bundles.