Perverse sheaves and intersection cohomology

Constructible complexes and the gluing of t-structures; the perverse t-structure, intermediate extensions and intersection complexes; Artin vanishing, nearby and vanishing cycles; weights and purity over finite fields and the decomposition theorem; small resolutions and Springer theory; equivariant perverse sheaves and the dictionary with D-modules and trace functions.

Twelve lessons with sixty worked exercise solutions.

The arguments use the geometric and coefficient hypotheses stated in each lesson. Appendices BZ–CB prove the geometric fibre comparison (BY.26) for bounded constructible complexes over a finite coefficient field of characteristic different from that of the finite or algebraically closed ground field; a finite extension of a finite ground field is allowed. Together with BO–BY, this gives the integral and rational realization statements under their stated coefficient hypotheses. The general course proofs remain incomplete: other canonical-appendix inputs and their transitive prerequisites, arbitrary-trait nearby cycles, several later comparison theorems and the general stack and Riemann–Hilbert statements require further proofs.

Before this course

• Constructible sheaves and extension by zero
• Sections with support and the localization triangle
• Truncation triangles and abelian hearts
• T-exact functors and adjoints between hearts

Lessons

1. Constructible complexes on algebraic varieties
2. Gluing t-structures
3. The perverse t-structure
4. Intermediate extensions and intersection complexes
5. Affine morphisms, Artin vanishing and perverse cohomology
6. Nearby and vanishing cycles
7. Weights, purity and semisimplicity over finite fields
8. The decomposition theorem
9. Semismall maps and small resolutions
10. Springer theory
11. Equivariant perverse sheaves and perverse sheaves on stacks
12. Perverse sheaves, D-modules and the function-sheaf dictionary

Download editable course sources

CC0 1.0.