# Summer school - Derived categories

Reception, lectures and short courses will be held at Amphi Pasteur Building 2 (see Practical informations - Campus map)

# Dennis Gaitsgory

### 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/

# Henning Krause

### 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.

Henning's course (texxed by P.Belmans)

# Alexander Kuznetsov

### 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.*

# Yukinobu Toda

### 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.

Exercises