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

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”