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Lefschetz decompositions and Categorical resolutions of singularities

Alexander Kuznetsov
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Abstract

Let $Y$ be a singular algebraic variety and let $\TY$ be a resolution of singularities of $Y$. Assume that the exceptional locus of $\TY$ over $Y$ is an irreducible divisor $\TZ$ in $\TY$. For every Lefschetz decomposition of $\TZ$ we construct a triangulated subcategory $\TD \subset \D^b(\TY)$ which gives a desingularization of $\D^b(Y)$. If the Lefschetz decomposition is generated by a vector bundle tilting over $Y$ then $\TD$ is a noncommutative resolution, and if the Lefschetz decomposition is rectangular, then $\TD$ is a crepant resolution.

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Alexander Kuznetsov. 2006-10-31. Lefschetz decompositions and Categorical resolutions of singularities. https://arxiv.org/abs/math/0609240

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