2025/2026

Fourier-Mukai Functors in Algebraic Geometry

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

When: 3, 4 module

Open to: students of one campus

Instructors: Alexander Pavlov

Language: English

Contact hours: 72

Course Syllabus

Abstract

"This is a specialized course in algebraic geometry. The main object of this course is the 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.  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 will 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. If time permits, we will discuss derived categories of K3 surfaces and/or abelian varieties."