Course by D. V. Pirozhkov "Derived categories of coherent sheaves" February 13–April 23, 2024, Steklov Mathematical Institute, Room 313 (8 Gubkina)
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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.
- Course overview. Triangulated categories.
- Derived categories of coherent sheaves. Serre duality.
- Bondal-Orlov reconstruction theorem.
- Fourier-Mukai transforms, equivalences of derived categories. Mukai duality.
- Rouquier dimension, Orlov’s conjecture about it.
Part II: semiorthogonal decompositions of derived categories.
- Exceptional objects, exceptional collections. *Semistable bundles on $\mathbb{P}^2$ (Drezet and
Le Potier) as the origin of this theory.
- Beilinson’s exceptional collection. Mutations of exceptional collections, dual
collections. *Exceptional collections on $\mathbb{P}^2$.
- *Kapranov’s exceptional collection on Grassmannians. Other collections on
Grassmannians (Fonarev, Kuznetsov-Polischuk).
- Semiorthogonal decompositions. Orlov’s decomposition for blow-ups and
projectivizations of vector bundles.
- Kawatani-Okawa rigidity theorem, examples of indecomposable derived
categories.
- 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
Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2022-265).
Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow Steklov International Mathematical Center |
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Course by D. V. Pirozhkov "Derived categories of coherent sheaves", February 13–April 23, 2024 |
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April 23, 2024 (Tue) |
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Lecture 10. Derived categories of coherent sheaves D. V. Pirozhkov April 23, 2024 18:00, Steklov Mathematical Institute, Room 313 (8 Gubkina)
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April 16, 2024 (Tue) |
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Lecture 9. Derived categories of coherent sheaves D. V. Pirozhkov April 16, 2024 18:00, Steklov Mathematical Institute, Room 313 (8 Gubkina)
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April 9, 2024 (Tue) |
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Lecture 8. Derived categories of coherent sheaves D. V. Pirozhkov April 9, 2024 18:00, Steklov Mathematical Institute, Room 313 (8 Gubkina)
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April 2, 2024 (Tue) |
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Lecture 7. Derived categories of coherent sheaves D. V. Pirozhkov April 2, 2024 18:00, Steklov Mathematical Institute, Room 313 (8 Gubkina)
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March 26, 2024 (Tue) |
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Lecture 6. Derived categories of coherent sheaves D. V. Pirozhkov March 26, 2024 18:00, Steklov Mathematical Institute, Room 313 (8 Gubkina)
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March 19, 2024 (Tue) |
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Lecture 5. Derived categories of coherent sheaves D. V. Pirozhkov March 19, 2024 18:00, Steklov Mathematical Institute, Room 313 (8 Gubkina)
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March 12, 2024 (Tue) |
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Lecture 4. Derived categories of coherent sheaves D. V. Pirozhkov March 12, 2024 18:00, Steklov Mathematical Institute, Room 313 (8 Gubkina)
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February 27, 2024 (Tue) |
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Lecture 3. Derived categories of coherent sheaves D. V. Pirozhkov February 27, 2024 18:00, Steklov Mathematical Institute, Room 313 (8 Gubkina)
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February 20, 2024 (Tue) |
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Lecture 2. Derived categories of coherent sheaves D. V. Pirozhkov February 20, 2024 18:00, Steklov Mathematical Institute, Room 313 (8 Gubkina)
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February 13, 2024 (Tue) |
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Lecture 1. Derived categories of coherent sheaves D. V. Pirozhkov February 13, 2024 18:00, Steklov Mathematical Institute, Room 313 (8 Gubkina)
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