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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..

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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).

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  • $\begingroup$ Could you please give a reference for $Perf(X)$ being the pullback of all the $Perf(\tilde{X}_k)$? $\endgroup$ – Evgeny Shinder Feb 25 at 8:02

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