Algebraic Geometry Reading Seminar on "Derived categories and GIT quotients".
Organizers: Sam DeHority, Junliang Shen.
Time: Thursday 4:15- 5:45. Room: KT 801.
The full notes of the seminar are kindly typed and shared by David Fang.
Assume that an algebraic variety X admits a (reasonably nice) action of a reductive group G. The classical Kirwan surjectivity theorem provides a surjective map from the equivariant cohomology of X to the GIT quotient, where the kernel of this map can be understood explicitly. This provides a powerful tool in calculating the cohomology of many moduli spaces in algebraic geometry.
The main purpose of this reading seminar is to understand a categorification of the Kirwan surjectivity. Roughly speaking, the equivariant bounded derived category of coherent sheaves on X admits a semi-orthogonal decomposition (SOD), and there is a component of the SOD, called the window category, describing precisely the bounded derived category of coherent sheaves of the GIT quotient.
This construction has many applications in algebraic geometry, representation theory, and mathematical physics. Some of them will be discussed later in the seminar.
The window category techniques have been introduced in:
[Se] Segal, Equivalence between GIT quotients of Landau-Ginzburg B-models, CMP 2011.
[HL] Halpern-Leistner, The derived category of a GIT quotient, JAMS 2015.
[BFK] Ballard, Favero, and Katzarkov, Variation of geometric invariant theory quotients and derived categories, Crelle 2019.
In this seminar, we will mainly follow [HL].
More references:
Derived categories.
[BO] Bondal and Orlov, Semiorthogonal decompositions for algebraic varieties, arxiv link
[Hu] Huybrechts, Fourier-Mukai transform in algebraic geometry, book link
[To] Torres, Windows in algebraic geometry and applications to moduli, thesis link
[Lec] Notes for Kuznetsov lectures on semiorthogonal decompositions and derived categories, notes link
Geometric Invariant Theory.
[Th] Thomas, Notes on GIT and symplectic reduction for bundles and varieties, arXiv link
[Lim] Lim, Notes on moduli spaces, note link
[Mu] Mukai, An introduction to invariants and moduli, book link
To participants: We assume basic knowledge about algebraic geometry (e.g. Hartshorne's book) and triangulated categories and derived functors (the participants are encouraged to skim through the first 2 chapters of [Hu]).
09.12. Organization meeting, and introductory talk by one of the organizers. (Junliang)
09.19. Derived category I. (Vlad)
Introduce (quickly) the derived category of coherent sheaves and derived functors following [Hu, Chapter 3];
Introduce Fourier-Mukai transform and some basic properties [Hu, Chapter 5]; state the key structural theorem --- Orlov Theorem [Hu, Theorem 5.14], without proof. Then state Gabriel's theorem [Hu, Corollary 5.24], and discuss the idea of its proof
09.26. Derived category II. (David Bai)
Introduce semi-orthogonal decomposition and exceptional collections following [Lecture 1, Lec]; see also the beginning of [Chapter 2, BO].
Work out the example of SOD (full exceptional collection) for P^1 in detail following [Lecture 2, Example, Lec] (method (b) for Step 2). Then sketch the proof for P^n following [Hu, Chapter 8.3].
If time permits, state the result (without proof) for Grassmaniann e.g. Kapranov. If there is extra time, do some other examples of SOD, e.g. cubic 4-folds, etc.
(No meeting on 10.03!)
10.10. Derived category III. (Sam)
Equivariant derived category of coherent sheaves.
October Recess.
10.24. Geometric Invariant Theory I. (David Bai)
A crash course on GIT: G-linearization of a line bundle, the definition of stable, semistable, and unstable locus, the Hilbert-Mumford criterion. This is roughly the material of [Lim, Sections 5.2 and 5.3]; a very good geometric introduction to GIT is given in [Th, Section 3].
(For this seminar, you are free to restrict yourself to linear reductive groups of type A, where explicit calculations are made more easily.)
10.31. Geometric Invariant Theory II. (Soumik)
Examples in GIT.
Do some simple examples in [Mu, Section 7.2]; then explain the Grassmannian case as in [Mu, Section 8.1].
Finally, sketch the case for vector bundles: slope stability = GIT stability.
11.07. Quantization theorem "i.e. quantization commutes with reduction". (Soumik)
The purpose is to explain [HL, Theorem 3.29].
See also [To, Theorem 1.4.1] and the remarks and examples after that; discuss [To, Example 1.4.4].
11.14. Derived Kirwan surjectivity I. (Kien)
State the main theorem, and discuss examples.
11.21. Derived Kirwan surjectivity II. (David Fang)
Discuss the proof.
Thanks giving!
12.05. Further research topics .... (Sam)