Non-commutative crepant resolutions: scenes from categorical geometry (Q2895438)

In [Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 47--56 (2002; Zbl 0996.18007)], \textit{A. Bondal} and \textit{D. Orlov} suggested the possibility of the existence of a categorical resolution of an algebraic variety \(X\). Such a resolution is a pair \((C,K)\) consisting of an abelian category \(C\) of finite homological dimension and of a thick subcategory \(K\subset D^b(C)\) such that the bounded derived category of coherent sheaves on \(X\) is \(D^b(C)/K\). In [The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3--8, 2002. Berlin: Springer. 749--770 (2004; Zbl 1082.14005)], \textit{M. Van den Bergh} defined a noncommutative crepant resolution, realizing the Bondal--Orlov categorical resolution, and in some cases proved the existence of such a resolution. The paper under review is an expository paper describing recent advances related to this topic.NEWLINENEWLINEVery recently, \textit{A. Kuznetsov} and \textit{V. A. Lunts} [``Categorical resolutions of irrational singularities'', \url{arXiv:1212.6170}] constructed a categorical resolution of any singularity over a field of characteristic zero. It is slightly weaker than the one conjectured by Bondal and Orlov.NEWLINENEWLINEFor the entire collection see [Zbl 1237.13005].