2021/2022
Research Seminar "Derived Categories of Coherent Sheaves"
Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
1, 2 module
Open to:
students of all HSE University campuses
Instructors:
Alexander Pavlov
Language:
English
ECTS credits:
3
Contact hours:
30
Course Syllabus
Abstract
This is a specialized course in algebraic geometry. The main object of this course is a derived category of coherent sheaves on a smooth projective variety. From a homological point of view it is an example of a triangulated category, satisfying a lot of additional nice properties. Derived categories of coherent sheaves have a long history: in 1960th they were used by A. Grothendieck and J.-L. Verdier to formulate and prove relative version of the duality theorem; in 1980th derived categories of coherent sheaves got new attention from the point of view of semi-orthogonal decompositions of triangulated categories; In 1990th they were used by M. Kontsevich to formulate homological mirror symmetry program. In the first part of the course we will review basics of triangulated categories, derived categories of abelian categories and derived functors, but we will focus mainly on applications of these methods to category of coherent sheaves on a smooth projective variety. We will prove basic facts about derived functors between derived categories of coherent shaves and work out several explicit examples of semi-orthogonal decompositions. We will find examples of geometric tilting: equivalence of the derived category of coherent sheaves and derived category of an associative algebra. We are also going to provide a criterion for an integral functor to be an equivalence. Then we will be able to show that a variety is completely determined by its derived category in the case of ample or anti-ample canonical bundle and, also, compute the group of derived automorphisms in these cases. We will find out that spherical objects allow constructing derived automorphisms of a non-geometric origin. PREREQUISITES: Students taking this course are assumed to be familiar with basics of algebraic geometry and homological algebra: categories, complexes and cohomology, injective and projective resolutions; varieties (projective, smooth), map of varieties (proper, projective, flat, smooth), line bundles and divisors, ampleness, vector bundles and coherent sheaves, Serre duality.
Learning Objectives
- To learn general homological methods of triangulated and derived categories.
- To gain knowledge of applications of homological methods in algebraic geometry.
- To understand proofs of fundamental theorems of the subject.
- To be able to apply homological methods in research.
Expected Learning Outcomes
- To become familiar with the notion of tilting bundle and how it gets an equivalence of the derived category of coherent sheaves and derived category of an associative algebra.
- To become skilled at using derived functors in algebraic geometry.
- To gain knowledge of triangulated categories.
- To know how spherical objects are used to construct derived automorphisms and be able to provide examples.
- To learn results on integral (Fourier-Mukai) functors and become skilled at applying these results to examples.
- To learn the proof of the reconstruction theorem for varieties with (anti)ample canonical class.
- To study derived categories of abelian categories as triangulated categories.
- To study proofs existence of the semi-orthogonal decompositions in the cases of projective space, quadric and Grassmanian.
- To understand the structure of the group of derived automorphisms for varieties with (anti)ample canonical class.
Interim Assessment
- 2021/2022 2nd module0.25 * Assignment 4 + 0.25 * Assignment 2 + 0.25 * Assignment 1 + 0.25 * Assignment 3
Bibliography
Recommended Core Bibliography
- Categories and sheaves, Kashiwara, M., 2006
- Methods of homological algebra, Gelfand, S. I., 2003
- Triangulated categories, Neeman, A., 2001
Recommended Additional Bibliography
- Weibel, C. A. (1994). An Introduction to Homological Algebra. Cambridge University Press.