Lefschetz decompositions and categorical resolutions of singularities

Abstract.

Let Y be a singular algebraic variety and let $\tilde Y$ be a resolution of singularities of Y. Assume that the exceptional locus of $\tilde Y$ over Y is an irreducible divisor $\tilde Z$ in $\tilde Y$. For every Lefschetz decomposition of the bounded derived category $\mathcal {D}^b(\tilde Z)$ of coherent sheaves on $\tilde Z$ we construct a triangulated subcategory ${\tilde{\mathcal{D}}}\subset{\mathcal{D}}^b({\tilde{Y}})$) which gives a desingularization of $\mathcal {D}^b(Y)$. If the Lefschetz decomposition is generated by a vector bundle tilting over Y then ${\tilde {\mathcal{D}}}$ is a noncommutative resolution, and if the Lefschetz decomposition is rectangular, then ${\tilde {\mathcal{D}}}$ is a crepant resolution.