Lefschetz decompositions and categorical resolutions of singularities

By Alexander Kuznetsov. 2006

118 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6d8f5780ed9173a838343dece0eb5bdda8d74c4a

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In Geometry and Topology

1. Alexander Kuznetsov
2. V. Lunts
3. K. Hori
4. P. Stellari
5. Emanuele Macrì
6. A. Neeman
7. Dmitri Orlov
8. Alexander Perry

In Mathematical Physics

1. V. Lunts
2. Alexander Kuznetsov
3. K. Hori
4. P. Stellari
5. A. Neeman
6. Emanuele Macrì
7. Alexander Perry
8. I. Smith

Citations in the corpus

• Relative singularity categories I: Auslander resolutions (2012, unknown): “Hence, the ringed space X = (X,A) is a non-commutative resolution of X and D(Coh(X)) is a categorical resolution in the spirit of works of Van den Bergh [87], Kuznetsov [60] and Lunts [65].”
• Categorical resolutions of irrational singularities (2012, background): “On the other hand, in [15] the definition of a categorical resolution (in the context of small categories) used here first appeared.”
• Semiorthogonal decompositions in algebraic geometry (2014, background): “of the derived pushforward functor Rf∗ : Db(coh(Y˜)) → Db(coh(Y)) is given by f!(F) = Lf∗(F)⊗O˜ Y (KY /Y˜ ), so for a geometric resolution crepancy and weak crepancy are equivalent. 12 Definition 3.6 ([K08c]). A categorical resolution (T ,π∗,π∗) of a scheme Y is strongly crepant if the relative Serre”
• Non-commutative resolutions of quotient singularities for reductive groups (2015, unknown): “Then the functor D(S)→ D( ) : N → M L ⊗S N is fully faithful and hence yields a categorical resolution of singularities of S in the sense of [33,35,42].”
• Exceptional collections for Grassmannians of isotropic lines (2005, background): “A Lefschetz exceptional collection is minimal if, roughly speaking, its first block is the minimal possible (see the precise definition in [7]).”