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).

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.