# Derived Categories of Varieties, Summer 2020

This summer we will have an online seminar for graduate students on derived categories of varieties leading up to the result that given a smooth proper variety over an algebraically closed field there are only a countable number of varieties derived equivalent to the given one. Our main text will be the corresponding chapter of the Stacks project.

Before being able to attend this seminar you need to contact me so I can add you to the mailing list where you'll get the zoom links for the talks. Everybody is expected to participate which will mean giving talks for graduate students and willing to answer questions for senior people. We want dumb questions by participants, so the seminar will have a limited size to encourage this and not everybody will be let in.

Preliminaries. We are going to assume you have an idea of what a derived category is, have an idea what the derived category D_{QCoh}(X) of quasi-coherent modules on a scheme is, have an idea what the bounded derived category D^b_{Coh}(X) of coherent modules on a Noetherian scheme is, have an idea what a perfect complex on a scheme is, have an idea what D_{perf}(X) ⊂ D^b_{Coh}(X) is if X is Noetherian. We are going to use as a black box all results on derived categories of schemes (including cohomology and base change).

Advice for the talks. In order to give a talk, strip the result you have to talk about from all generalities and try to explain the proof (using anything in the black box above) of the most straightforward version of the result you can think of.

Preliminary list of talks:

1. Representability and existence of adjoints Prove Bondal-Van den Bergh's theorem that a finite type cohomological functor on D(X) is representable. See Theorem 0FYH. The main application will be the existence of adjoints as in Lemma 0FYN. Link to first set of lecture notes.
2. Fourier-Mukai functors Talk about Fourier-Mukai functors, see Section 0FYP Please show that pullback by a morphism is a Fourier-Mukai functor and tensoring by an invertible module is too. Also, please explain the category of Fourier-Mukai functors in Section 0G0F. If you have time, explain the original Mukai story about an abelian variety and its dual. Alternatively, you can talk a little bit about Orlov's paper on Fourier-Mukai functors and motives to see how Fourier-Mukai functors give you interesting geometric information.
3. Strong generators and boundedness Explain why D_{perf}(X) has a strong generator for X smooth and proper, see Lemma 0FZ6. This is explained beautifully and in much greater generality in the paper by Bondal and Van den Bergh. Explain moreover why this means functors emanating from D_{perf}(X) are "bounded", see Lemma 0FZ8. This is explained in Orlov's paper on K3s. See Section 0FYZ
4. Gabriel type results Discuss the much softer material in both Section 0FZA and Section 0FZK. We won't use the last lemma on Gabriel's result that Coh(X) determines X, but it might be a fun endpoint of the discussion in this lecture.
5. Sibling functors Cover the material in Section 0FZS and then state and prove Lemma 0G00. We should maybe add a little bit more to this talk, but I am not sure yet.
6. Orlov's argument Given a fully faithful functor F : D_{perf}(X) → D_{perf}(Y) explain how to get the Fourier-Mukai functor G : D_{perf}(X) produced in Section 0G07. Please explain after you've produced G for what sheaves ℐ on X we get F(ℐ) isomorphic to G(ℐ). (Vanishing cohomology.) This is well explained in Orlov-K3 and in Ballard's writeup allthough our approach is sligtly different.
7. Finish Orlov's result Put everything together to get Tag 0G0B Tag 0G0C Tag 0G0D Tag 0G0E This takes some doing because of the labyrinthine structure of the argument going from the result in the previous lecture to the existence of sibling. Try to simplify it if you can!
8. No deformations Explain why you cannot deform an X whilst keeping the derived category the same, see Section 0G0M This includes defining what is "the kernel of a relative Fourier-Mukai equivalence", but please refrain from proving lemmas about this because this should be clear from our discussion of Fourier-Mukai functors
9. Countability Explain the proof of the theorem by Anel and Toen that given a smooth proper X over an algebraically closed field k there are at most countably many isomorphism classes of Y over k which are derived equivalent to X. See Section 0G0Z