# Derived category of singular varieties

Let $$X$$ be a projective variety with only normal crossing singularity. Is there a description of the derived category or the category of perfect complexes? What about the existence of semiorthogonal decomposition?

Please provide a reference if this kind of study exists in the literature..

Let $$\tilde{X}_k$$ be the normalization of the closed $$k$$-codimension stratum, so $$\tilde{X}_0$$ is the normalization of $$X$$. Then there is a diagram of pullback functors between the categories $$\text{Perf}(\tilde{X}_k),$$ such that $$\text{Perf}(X)$$ is the pullback of this diagram in the $$\infty$$-category of derived categories (for example perfect derived sheaves on the cross $$xy = 0$$ are pairs of sheaves on $$\mathbb{A}^1$$ with fiber at $$0$$ identified). You can then perhaps use this paper https://arxiv.org/pdf/1901.01257.pdf by Scherotzke-Sibilla-Talpo to glue semiorthogonal decompositions on the $$\text{Perf}(\tilde{X}_i)$$ to a semiorthogonal decomposition on $$X$$.
Of course if you're just interested in some (unbounded) quasicoherent derived category, like for example $$D^-\text{QCoh}$$ then the filtration itself gives a semiorthogonal decomposition (define the filtered piece $$D^-\text{QCoh}_{\le k}$$ to be the category of sheaves supported on the closed $$k$$-stratum. Then the semi-orthogonal pieces associated to this filtration will be categories $$D^-\text{QCoh}_k$$ generated by pushforwards of sheaves from the open $$k$$-dimensional stratum).
• Could you please give a reference for $Perf(X)$ being the pullback of all the $Perf(\tilde{X}_k)$? – Evgeny Shinder Feb 25 at 8:02