Advanced Topics in Algebraic Geometry (V5A3): Winter term 21/22
In this class we will explore categorical methods to approach the geometry of
algebraic varieties. More precisely, for a (smooth, projective) variety X will study the
bounded derived category D^b(Coh(X)) of coherent sheaves on X. This has become a central objects of study over the last twenty years. Derived categories are used to link different types of varieties, to produce cycles, to construct moduli spaces, to encode geometric properties (like rationality), etc. The emphasis will be on
the geometric aspects rather than on the abstract language of categories (e.g. will avoid the language of higher categories, A_infty categories, etc.). Special focus
will be on derived aspects of (cubic) hypersurfaces.
Here are a few topics that I am planning to cover.
Abelian categories Coh(X), Coh(X,a) of (twisted) coherent sheaves, Gabriel's theorem
Derived and triangulated categories, semi-orthogonal decompositions, left & right mutations, exceptional objects, blow-up formula for derived categories
Derived functors and Fourier--Mukai functors, derived equivalences of varieties, Bondal-Orlov theorem
Generators in triangulated categories, Rouquier dimension, Orlov spectrum
Fano variety of lines on cubic hypersurfaces
Homological projective duality
Stability conditions, bounded t-structures, hearts, torsion theories, tilting
Prerequisits: Solid knowledge of algebraic geometry (e.g. as covered by my class last year, roughly the content of Hartshorne's book. Some familiarity with homological methods will be useful. The construction of the derived category will be recalled (depending on the audience).
A. Bondal & D. Orlov: Reconstruction of a variety from the derived category and groups of autoequivalences. 2001
A. Bondal and M. van den Bergh: Generators and representability of functors in commutative and noncommutative geometry. 2003
S. Gelfand and Y. Manin. Methods of homological algebra. Springer Monographs in Mathematics. 2nd edition 2003
D. Huybrechts: Fourier-Mukai transforms in Algebraic Geometry. Oxford Mathematical Monographs. 2006
D. Huybrechts: Lectures on cubic hypersurfaces. in preparation.
A. Kuznetsov: various articles
D. Orlov: Remarks on generators and dimensions of triangulated categories 2008
R. Rouquier: Dimensions of triangulated categories 2003
Currently the class is planned to take place in person. Monday & Friday 2-4pm, SR 1.008
Please register for this class on ecampus. The password is FourierMukai. The class starts Oct 15.