Summer school - Derived categories
Reception, lectures and short courses will be held at Amphi Pasteur Building 2 (see Practical informations - Campus map)
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
Some materials for these lectures will be posted at: http://www.math.harvard.edu/~gaitsgde/Nantes14/
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.