École d'été - Catégories dérivées
Lie theory from the point of view of derived algebraic geometry
- (Lecture 1: The basics)
(a) Deformation theory
(b) The notion of *inf-scheme*
(c) Ind-coherent sheaves on inf-schemes
- (Lecture 2: Lie algebras and group inf-schemes ) (a) Formal moduli problems à la Lurie
(b) Review of Quillen duality
(c) The exponential construction
- (Lecture 3: Lie algebroids) (a) The inertia group
(b) The notion of Lie algebroid in derived algebraic geometry
(c) Relation the classical notion of Lie algebroid
- (Lecture 4: applications of Lie algebroids) (a) Deformation to the normal bundle
(b) The notion of n-th infinitesimal neighborhood
(c) Relation to the BPW filtration
Stratification of triangulated categories
The aim of these lectures is to explain how derived categories arising in algebra and geometry can be
stratified. A prototypical result is the Hopkins-Neeman classification of localising subcategories of the
derived category of a commutative noetherian ring. This will be discussed in some detail. A basic tool are
local cohomology functors for triangulated categories which are defined with respect to a central ring
action. This leads to a notion of cohomological support which is inspired by the study of support varieties in
modular representation theory. Further concepts to be disussed are: local-global principles, tensor
triangulated structures, homotopy categories of injectives, and exceptional sequences. A useful reference
is: D.J. Benson, S.B. Iyengar, H. Krause, Representations of finite groups: Local cohomology and support,
Oberwolfach Seminars 43, Birkhäuser Verlag, 2012, 111 pp.
Derived categories of cubic 4-folds
- (Topic 1) An overview of semiorthogonal decompositions; derived categories of cubic hypersurfaces;
the Serre functor of their nontrivial components.
- (Topic 2 ) (a) Symplectic structure on moduli spaces of objects; The symplectic structure of the Fano scheme of lines.
- (Topic 3) Derived categories of cubic fourfolds containing a plane.
- (Topic 4) Derived categories of Pfaffian cubics.
- (Topic 5) Derived categories and the Fano scheme of lines.
Stability conditions and Donaldson-Thomas invariants
- (Topic 1) Bridgeland stability conditions
The notion of stability conditions on triangulated categories was introduced by Bridgeland in 2002, as a mathematical framework of Douglas’s Pi-stability. I will give an introduction to Bridgeland stability conditions and explain how they are related to mirror symmetry and birational geometry.
- (Topic 2 ) Donaldson-Thomas invariants.
The Donaldson-Thomas invariants are invariants counting stable coherent sheaves on Calabi-Yau 3-folds. They were introduced by Thomas in 1998, and later generalized by Joyce-Song, Kontsevich-Soibelman. I will give an introduction to Donaldson-Thomas invariants and explain some results and conjectures on them. I will explain the role of Bridgeland stability conditions for these problems.