Course Identification

Some familiarity with Category theory and Sheaves Theory

In this course I will describe the theory of perverse sheaves and present some applications of this theory. I will start with reminder of basic notions of sheaf theory, explain why in addition to abelian category of sheaves one has to study its derived category.

If X is a complex algebraic variety we can consider subcategories of constructible sheaves and complexes. They contain essentially all the geometrically interesting objects. I will explain why these subcategories are preserved by natural functors. These categories can be related to some categories of D-modules. This relation suggest a definition of some new abelian category - category Perv of perverse sheaves. The formal definition of this category is rather involved, but its relation with D-modules shows that this is the "correct" category to study. I will describe main properties of this category. A. Weil suggested that the usual theory of cohomologies of complex algebraic varieties can be generalized for varieties over other fields. This project was realized by A. Grothendieck who introduced etale topology and defined sheaves in this more general case. It turns out that in this case one can also consider the category of perverse sheaves. I will describe main properties of these notions. In case of varieties over finite field this can be generalized by adding additional structure - weight filtration. I will describe main results by P.Deligne about this theory.

I. Generalities on sheaves

1. General properties of sheaves. Standard functors.
2. Derived categories of sheaves.
3. Functors between derived categories of sheaves:. (i) Push forward and ! push forward; (ii) Pull back and ! pull back; (iii) Verdier duality
4. Standard triangle

II. Sheaves on complex algebraic varieties

1. Local systems and their properties.
2. Perverse sheaves.
3. Relation with D-modules
4. Functors on perverse sheaves.
5. Properites of the category Perv of perverse sheaves
6. Classification of irreducible perverse sheaves.

III. Functors of nearby cycles and vanishing cycles

IV. Weil conjectures

1. Etale topology and comparison theorems.
2. -adic contructible sheaves. Derived category of l-adic sheaves.
3. Functors.

V. Sheaves over varieties over finite fields.

1. Frobenius.
2. Weil sheaves and Weil complexes.
3. Mixed complexes. Functors. Sheaves to functions correspondence.
4. Weight filtration.
5. Main results about weight filtration.

VI. Filtration on perverse sheaves

1. Purity of irreducible perverse sheaves.
2. Decomposition theorem for pure sheaves.
3. Decomposition theorem for push forward with respect to proper mor-phisms
4. Constructible sheaves and complexes.
5. Functors between categories of constructible complexes.

II. t-structures

1. t -structures on triangulated categories. Heart of t-structure.
2. Perverse t-structure
3. Case of small and semi-small proper morphisms.
4. Nearby cycles, perversity and ltration.

VII. Applications

1. Localization theorem and Kazhdan-Lusztig conjectures.
2. Character sheaves
3. Springer resolution

VIII. Equivariant derived categories

1. Equivariant derived category and equivariant perverse sheaves.
2. Applications to toric varieties

Upon successful completion of this course students should be able to:

1. Describe the theory of perverse sheaves and present some applications of this theory
2. Describe main results by P.Deligne about this theory.
3. Discuss applications of perverse sheaves and weight filtration to several problems that arise in Representation Theory and Algebraic Geometry.

1. B. Iversen, "Cohomologies of sheaves", Springer-Verlag
2. A.A. Beilinson, J. Bernstein, P. Deligne "Faisceux pervers". Asterisque 100, 3-171 (1983)
3. R. Kiehl, R. Weissauer  "Weil Conjectures, Perverse Sheaves and l-adic Fourier Transform"