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.