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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?

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    $\begingroup$ You'd better call it "(triangulated) category of singularities". Not to confuse with smoothness. $\endgroup$
    – Anton Fonarev
    Commented Feb 8, 2014 at 7:49
  • $\begingroup$ 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. $\endgroup$
    – user36931
    Commented Feb 8, 2014 at 13:07
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    $\begingroup$ Also Abouzaid, Auroux, Efimov, Katzarkov, Orlov make a computation in some explicit cases for their joint paper. $\endgroup$
    – user36931
    Commented Feb 8, 2014 at 13:08
  • $\begingroup$ @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. $\endgroup$
    – math no more
    Commented Feb 8, 2014 at 23:08
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    $\begingroup$ arxiv.org/abs/1012.5282 and references therein $\endgroup$
    – user36931
    Commented Feb 9, 2014 at 3:00

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A student of Orlov is working on one case. He will give a talk in two weeks in Padova, here is the abstract:

  • Oleksandr Kravets (Moscow Higher School of Economics, Russia), Exceptional collections in categories of singularities of 3-dimensional Landau-Ginzburg models.

Abstract: Derived categories of singularities measure how far is an algebraic variety from being smooth. They appear on the algebraic side of the Homological Mirror Symmetry (HMS) for the Landau-Ginzburg models. The talk will be dedicated to the case when the superpotential is given by the so called invertible polynomial. In the case of small dimensions, I will give the description of mentioned categories in terms of full strongly exceptional collections of good kind. I will also recall several conjectures concerning HMS for the current case and will show how the constructed collections help to assert

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