Derived equivalence for the simple flop of type $D_5$

TL;DR

The paper proves that every simple flop of type $D_5$, resolved by two blowups with exceptional divisor $OG(4,V_{10})$, induces a derived equivalence $D^b(X_+)\cong D^b(X_-)$. The approach hinges on a prolonged sequence of mutations of exceptional objects, organized via a chessboard formalism, and relies on careful cohomology computations for homogeneous bundles using the Borel–Weil–Bott framework. It also connects these flop phenomena to Calabi–Yau geometry: Calabi–Yau fivefolds of type $D_5$ appear as zero loci in the roof construction, and the authors show non-birational but derived-equivalent CY5 pairs (the so-called double mirrors), extending the method to Calabi–Yau fibrations over a base. The results provide new evidence for the DK conjecture of Bondal–Orlov and Kawamata and extend the mutation-based strategy to relative settings, highlighting a unified mechanism for derived equivalences in both flops and CY5 double-mirror constructions.

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.

Figures (2)

• Figure 1: SOD of $\mathbb S_+$
• Figure 2: SOD of $^{\perp}D^b(X_+)$

Theorems & Definitions (49)

• Conjecture 1.1: DK conjecture bondalorlovkawamata
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Example 2.3
• Example 2.4
• Remark 2.5
• Lemma 2.6
• proof