# For what varieties do we have results on the category of singularities?

Let $X$ be a singular variety. Define the (triangulated) category of singularities (as in Orlov's paper) as the Verdier quotient of the derived category of coherent sheaves on $X$ modulo the full subcategory of perfect complexes.

For example, there is a quiver description in the case of ADE surface singularities: http://arxiv.org/abs/math/0511155

Are there any other cases do we have results for the category of singularities? In particular, for higher codimension varieties?

• You'd better call it "(triangulated) category of singularities". Not to confuse with smoothness. Feb 8, 2014 at 7:49
• There are of course the cases of Landau-Ginzburg Calabi-Yau (and not Calabi-Yau!) correspondence See the papers of Orlov, Segal, Shipman, Itsik,... where the category of singularities can be related to the usual derived category of the critical locus. Feb 8, 2014 at 13:07
• Also Abouzaid, Auroux, Efimov, Katzarkov, Orlov make a computation in some explicit cases for their joint paper. Feb 8, 2014 at 13:08
• @user36931 Thanks for the reference. Do you know a title for any of the papers by Orlov, Segal, Shipman, Itsik? I'm having trouble finding them, possibly because I'm unfamiliar with what I should be looking for. Feb 8, 2014 at 23:08
• arxiv.org/abs/1012.5282 and references therein Feb 9, 2014 at 3:00