D-equivalence conjecture for birational Calabi–Yau varieties

Establish that any two smooth projective birational Calabi–Yau varieties X and X′ have equivalent bounded derived categories, D^b(X) ≃ D^b(X′).

Background

The paper recalls the D-equivalence conjecture of Bondal–Orlov and Kawamata, predicting derived equivalence for smooth projective birational Calabi–Yau varieties. The authors prove the conjecture for hyperkähler varieties of OG10-type, but the general conjecture remains unresolved.

The paper’s construction of twisted hyperholomorphic sheaves supplies the derived equivalences needed for the OG10 case and extends earlier results for hyperkähler manifolds of K-type.

References

If X and X' are smooth projective birational Calabi--Yau varieties, then there is an equivalence of bounded derived categories Db(X) \simeq Db(X').

— Equivalences via twisted hyperholomorphic sheaves from transverse Lagrangian fibrations  (2608.13403 - Hartlieb et al., 13 Aug 2026) in Conjecture 1.2, Section 1, Introduction

Along with the results for $K3{[n]}$-type and OG10 type, this leaves generalized Kummer type as the only remaining case among the currently known deformation types of irreducible holomorphic symplectic manifolds where the conjecture has not yet been fully established.

— A Strong Global Torelli theorem for OG6 type varieties  (2609.28939 - Ling, 24 Sep 2026) in Introduction

Note that this result is compatible with the conjecture that the existence of flops implies derived equivalence *{Conjecture in p.40} since if we have a floppable curve $C \subset X$, then $\omega_X|_C$ is torsion and hence $\omega_X$ is somewhere torsion.

— Bondal--Orlov reconstruction for tame stacks with trivial generic stabilizer  (2609.28867 - Ito et al., 24 Sep 2026) in Remark immediately following Theorem 2.1, Section 2.2

Optimistically, one should expect that the twisted variant of Theorem 0.3 still holds, and that a similar proof will suffice. As the proof will show, this is related to the question of whether a Brauer class admits a B-field lift which is isotropic with respect to the BBF form.

— Equivalences via twisted hyperholomorphic sheaves from transverse Lagrangian fibrations  (2608.13403 - Hartlieb et al., 13 Aug 2026) in Remark immediately following Theorem 6.1, Section 6, Subsection “The D-equivalence conjecture for hyperkähler manifolds of OG10-type”