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Given my limited knowledge on derived categories, all the results on derived categories of complex of bounded sheaves are build upon smooth varieties, and people literally avoid singular case (as in the case of Bridgeland and Maciocia's proof of Fourier-Mukai transforms on abelian fiberation etc.)

I want to know

(1) What are the main difficulties working on singular and noncomplete varieties? (the resolution is not finite? the dimension of cohomology group is not finite? Or no spectral sequences? Then what's the exact trouble?)

(2) What is the current knowledge of derived categories on arbitrary varieties?

My question might be too big to be answered precisely -- any references are extremely welcome!!!

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    $\begingroup$ There's even something called the derived category of singularities, which is only non-trivial for singular varieties, so I wouldn't say that people avoid the singular case. $\endgroup$ – Fernando Muro Dec 4 '13 at 21:48
  • $\begingroup$ That is really interesting! Could you point out the references? $\endgroup$ – Li Yutong Dec 4 '13 at 22:53
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    $\begingroup$ @LiYutong D.Orlov, "Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities" $\endgroup$ – Anton Fonarev Dec 5 '13 at 11:48
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Probably the main reason people avoid singular varieties is because of boundedness. Arguments using induction and, as you say, spectral sequences need boundedness of the complexes in play in one direction or the other (and sometimes both). Related to this, is the failure of Serre duality, which lies at the heart of many derived arguments. (the fact that the Serre functor commutes with any equivalence is a very nice tool to throw around)

In the smooth and proper case the three main functors to build integral transforms (pushforward, tensor, pullback) all preserve boundedness. In the just proper case pushforward is still fine. Pullback from $X \times Y \to X$ is of course not the issue (unless you're doing some crazy relative situation) as the projection map is flat. So the real problem is in taking tensor products. If the kernel is not perfect then tensoring with it will escape to the bounded-above derived category.

So if you planned to run an argument by induction on the length of the complex which started at the bottom you're done for. For example, having nice formulae for the adjoints of an integral transform is no longer automatic as one cannot blindly apply Serre duality.

The derived category of singularities Fernando mentions is a bit of a silly name and refers to work of Orlov (originally, later extended by many many people). The point is that the category of perfect complexes sits inside the bounded derived category and one can take the (Verdier) quotient. You call this the derived category of singuarities. (for example, the structure sheaf of a singular point is not perfect, as it does not admit a finite resolution by vector bundles, so it will "survive" this quotienting procedure) In turn this category of singularities is often related to categories of matrix factorisations.

Perhaps I should mention that in the direction of Serre duality for singular varieties there is a whole school that thinks one should replace the unbounded derived category of quasi-coherent sheaves with the (bigger!) category of Ind-coherent sheaves. Vaguely (ie at the level of my understanding) perfect complexes form the compact objects of $D(QCoh(X))$. As on a singular varieties there are more objects we care about (eg coherent sheaves!) one would like a category where the whole $D^b(Coh(X))$ is the subcategory of compact objects. This wild Ind-Coh is tailored to do just that. I'm still not sure what formal obstacles this improves upon. I think they were first introduced by Krause, with the very concrete description of being the homotopy category of unbounded complexes of injective quasi-coherent sheaves. I vaguely remember there being some work of Murfet relating this to matrix facotrisations, but I might be mistaken.

EDIT: Another example which is very relevant. For a blow up $Y \to X$, there is a nice formula for $D(Y)$ in terms of $D(X)$ and $D$ of the exceptional locus. This holds, of course, if everything (including the centre) is smooth. As soon as you blow something singular this is no longer true. It was mentioned somewhere on this website by the user Sasha (Kuznetsov?) that there are examples where $D(Y)$ is actually indecomposable over $D(X)$, which is very odd and disappointing.

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  • $\begingroup$ Why Serre duality does not hold, I thought Cohen–Macaulay Scheme is enough? Besides, by the "unboundness" , do you mean there is no bounded injective resolution for a coherent sheaf (also for bounded complex)? Then the bounded above category should work? Is there any birational geometry idea used in remedy this problem instead of working on the category part? $\endgroup$ – Li Yutong Dec 5 '13 at 1:52
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    $\begingroup$ I wish I knew what birational geometers thought the correct derived category were. There is a difference between having a dualizing complex and having a Serre functor (which is what I should have said). I've always been confused by this. Let's define $D_X = s^!k$ to be the !-pullback of the ground field. The functor $\text{Hom}(-,D_X)$ is an autoequivalence of the bounded derived category. $\endgroup$ – John Salvatierrez Dec 5 '13 at 10:38
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    $\begingroup$ On the other hand a Serre functor $S$ is the datum of Serre duality, ie functorial isomorphisms $\text{Ext}^i(A,B) = \text{Ext}^{n-i}(B, S(A)}$. If you take $S$ to be $(-)\otimes D_X$ this does not work unless $X$ is smooth. This is not because the dualizing object is not good, but rather that not every complex is perfect, and so the formula with exts will run into trouble. $\endgroup$ – John Salvatierrez Dec 5 '13 at 10:40

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