Progress on Bondal–Orlov derived equivalence conjecture

Question

In their 1995 paper, Bondal and Orlov posed the following conjecture:

If two smooth $n$-dimensional varieties $X$ and $Y$ are related by a flop, then their bounded derived categories of coherent sheaves are equivalent as triangulated categories, i.e. we have $D^b(\mathsf{Coh}(X)) \cong D^b(\mathsf{Coh}(Y))$.

Selected 'particular' cases:

Tom Bridgeland proved that this holds in the $n=3$ case by showing that birational smooth projective Calabi-Yau threefolds are derived equivalent. This follows from the fact that any birational transformation between two $3$-dimensional Calabi–Yau varieties can be decomposed into a sequence of flops.

Ed Segal has also constructed an example in the $n=5$ case, and Daniel Halpern-Leistner has sketched a proof of the conjecture for the case of Calabi-Yau manifolds which are birationally equivalent to a moduli space of Gieseker semistable coherent sheaves (of some fixed primitive Mukai vector) on a K3 surface. As Sasha mentions in the comments, Yujiro Kawamata has also proved the conjecture in the toric case.

There are also other cases in which the conjecture holds which I have not mentioned - thank you in advance for any comments/answers highlighting these.

The general case?

I have heard however that a proof of this conjecture in general seems rather far off at this moment in time. I am interested in whether any progress has been made with regards to the general case, and what approach/techniques may be involved in a potential proof?

Answer 1

A lot of work has been done to prove the Bondal-Orlov conjecture for (stratified)-Mukai flops. To quote a few names : Namikawa (Mukai flops), Kawamata (one stratified Mukai flop), Cautis and collab (all stratified Mukai flops type A), Halpern-Leistner (reproves Cautis and collab results with other techniques). In all these examples however, the bases of the contrated locus are "the same" on the two sides of the flop.

The example exploited by Segal comes after all these examples. Abuaf indeed noticed it is an interesting example because the bases of the contracted locus are very different on the two sides of the flop. More such examples : https://arxiv.org/abs/1812.10688

Note also that Wierzba and Wisniewski proved that any two symplectic resolutions of the same symplectic singular four-dimensional variety are connected by a chain of Mukai flops. In particular, the Bondal-Orlov conjectures is true for symplectic flops in dimension 4 since we know it is true for Mukai-flops in dimension of type A.