Exceptional collections in algebraic geometry
Time and Place:
Lectures: Monday, 10-12 c.t., Room B 252.Exam:
The exam will take place on Thursday, February 13, from 10:00 until 12:00 in room B252. No materials are permitted in the exam. Please ensure you bring with you an appropriate identity document with photo. If you are considering taking the exam, please register by sending me (ananyevs@math.lmu.de) a short email with your name and Matrikelnr, and also register at LSF.The retake exam will take place on Monday, April 07, from 10:00 until 12:00 in room B431. No materials are permitted in the exam. Please ensure you bring with you an appropriate identity document with photo. If you are considering taking the retake exam, please register by sending me (ananyevs@math.lmu.de) a short email with your name and Matrikelnr.
Abstract:
The derived category of coherent sheaves on a smooth projective algebraic variety is a very powerful cohomological invariant of the variety containing the information about vector bundles, algebraic cycles, Hochschild homology and much more, and reflecting a lot of geometrical information. These categories originally appeared as a formal framework for derived functors (Ext, Tor, higher direct images, etc.) and duality, but later they drew a substantial attention when they were seen to be deeply related to the geometry of the variety. The derived categories of some varieties admit a particularly simple description via the so-called full exceptional collections (these are collections of objects in the category possessing some nice properties), providing an equivalence of the derived categories of the varieties in question with the derived categories of modules over some non-commutative algebras and making a bridge between geometry and representation theory. Conjecturally, such full exceptional collections exist on every projective homogeneous variety G/P with G being a linear algebraic group over complex numbers, but this is currently proved only in some particular cases.The course is supposed to be an introduction to this beautiful subject. We will start with the basics introducing the derived categories of modules and coherent sheaves, and proceed to an overview of some central results, including Bondal-Orlov reconstruction theorem, the notion of Fourier-Mukai transforms, construction of phantom categories, and construction of full exceptional collections on various homogeneous varieties, including projective spaces (by Belinson), Grassmannians and quadrics (Kapranov), and others.
Prerequisites:
This course will be of interest to Master students majoring in Algebra. I will assume that the students have some familiarity with Algebraic Geometry and Homological Algebra.The course will be taught in English.
References:
[1] Andrei Caldararu. Derived categories of sheaves: a skimming, arXiv:0501094[2] Daniel Huybrechts. Fourier-Mukai transforms in algebraic geometry, Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2006.
[3] Alexander Kuznetsov. Semiorthogonal decompositions in algebraic geometry, in Proceedings of the International Congress of Mathematicians. Vol. II, Seoul, 2014, pp. 635–660
[4] Anton Fonarev. Derived categories of Grassmannians: a survey, arXiv:2407.07455
Notes:
Handwritten notes for the coursePh.D. Alexey Ananyevskiy