Bondal–Orlov localization conjecture

Let $X$ be a singular variety and let

$$\pi:\widetilde{X}\to X$$

be a resolution of rational singularities. Write $\textbf{D}^{b}(X)$ and $\textbf{D}^{b}(\widetilde{X})$ for their bounded derived categories of coherent sheaves, and let $\ker\pi_{*}$ denote the kernel of the derived pushforward. Bondal–Orlov localization conjecture. The functor induced by $\pi_{*}$ should be an equivalence

$$\widetilde{\pi}_{*}:\textbf{D}^{b}(\widetilde{X})/\ker\pi_{*}\xrightarrow{\sim}\textbf{D}^{b}(X).$$

This conjecture proposes that the derived category of a singular variety can be recovered from a resolution by quotienting out the objects annihilated by pushforward. The source gives no resolution status, so the conjecture is recorded as open.