Derived category of coherent sheaves is a very large, but important invariant of an algebraic variety. Originally derived categories were mostly used as an auxiliary technical object that simplifies some aspects of working with derived functors and Grothendieck duality, but later people started studying derived categories as independently interesting mathematical objects. The properties of derived categories of coherent sheaves depend strongly on the geometry of the variety, but for some varieties the structure of the derived category has surprisingly simple algebraic description. We will discuss some aspects of the theory of derived categories. Prerequisites: you should know basic algebraic geometry (for example, you need to know what a coherent sheaf on an algebraic variety is), and some acquaintance with spectral sequences is desirable.  

Program:

Part I: derived category as an invariant.

1. Course overview. Triangulated categories.
2. Derived categories of coherent sheaves. Serre duality.
3. Bondal-Orlov reconstruction theorem.
4. Fourier-Mukai transforms, equivalences of derived categories. Mukai duality.
5. Rouquier dimension, Orlov’s conjecture about it.

Part II: semiorthogonal decompositions of derived categories.

6. Exceptional objects, exceptional collections. *Semistable bundles on $\mathbb{P}^2$ (Drezet and Le Potier) as the origin of this theory.
7. Beilinson’s exceptional collection. Mutations of exceptional collections, dual collections. *Exceptional collections on $\mathbb{P}^2$.
8. *Kapranov’s exceptional collection on Grassmannians. Other collections on Grassmannians (Fonarev, Kuznetsov-Polischuk).
9. Semiorthogonal decompositions. Orlov’s decomposition for blow-ups and projectivizations of vector bundles.
10. Kawatani-Okawa rigidity theorem, examples of indecomposable derived categories.
11. Cubic fourfold, $K_3$ categories, Kuznetsov’s rationality conjecture.

* - topics could be omitted depending on the students’ preferences and the course progress.

Lecture notes (in Russian)

Lecturer
Pirozhkov Dmitry Vladimirovich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center