Derived Categories of Varieties, Summer 2020
Professor A.J. de Jong,
Columbia university,
Department of Mathematics.
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:
- 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.
- 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.
- 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
- 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.
- 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.
- 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.
- 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!
- 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
- 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