The Bondal-Orlov Localization Conjecture Holds for Threefolds

Abstract

Let $X$ be a noetherian scheme with the resolution property, and let $p:Y\to X$ be a projective morphism. Suppose that $R^i{p}_{\ast }=0$ for $i>2$ and that ${\mathcal{O}}_X⟶R{p}_{\ast }{\mathcal{O}}_Y$ is an isomorphism. We show that derived pushforward induces an equivalence \[ D^{b}(Y)/\operatorname{Ker}(Rp_{*})\simeq D^{b}(X). \] As an application, we prove a characteristic-free form of the Bondal--Orlov localization conjecture for quasi-projective threefolds.