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

• Kuznetsov component

• Fano variety of lines on cubic hypersurfaces

• Homological projective duality

• Balmer spectrum

• Stability conditions, bounded t-structures, hearts, torsion theories, tilting

• Matrix factorization

• Fano visitors

• 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).

• References:
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.