Bondal–Orlov–Kawamata crepant-resolution derived-equivalence conjecture

Let $X$ be a normal algebraic variety with Gorenstein singularities, and let

$$\pi:Y_i\rightarrow X\qquad (i=1,2)$$

be two crepant resolutions by schemes or Deligne–Mumford stacks. An equivalence of triangulated categories is said to be linear over $X$ when it is compatible with the maps to $X$. Bondal–Orlov–Kawamata's conjecture. There is an equivalence

$$F:\operatorname{D}^{\operatorname{b}}(\operatorname{coh} Y_1)\simeq \operatorname{D}^{\operatorname{b}}(\operatorname{coh} Y_2)$$

linear over $X$. This conjecture expresses the expectation that crepant resolutions have the same categorical geometry; the source presents it as central motivation for non-commutative resolutions, without recording a general resolution.