Derived equivalence for the simple flop of type $D_5$

TL;DR

This paper proves derived equivalences for simple flops of type D_5 and related Calabi--Yau varieties, supporting the DK conjecture and extending the understanding of derived categories in algebraic geometry.

Contribution

It establishes derived equivalences for a class of simple flops of type D_5 and related Calabi--Yau varieties, using mutation of exceptional objects.

Findings

Derived equivalence for simple flop of type D_5.

Derived equivalence for certain non-birational Calabi--Yau fivefolds.

Extension to Calabi--Yau fibrations as zero loci in Grassmann bundles.

Abstract

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.