1. Introduction and Contents

We will introduce the formalism of constructible sheaves on complex algebraic varieties, which is the basis for the cohomological study of the geometry and topology of algebraic varieties.  In the first part, we will cover derived categories of sheaves, the six functors, constructibility, and Verdier duality.  In the second part, we will study nearby and vanishing cycles sheaves, which are used to analyze the topology of degenerating families.

(1)    Derived categories of sheaves on topological spaces.
(2)    The six functors: direct and inverse image functors, their compactly supported variants, and the tensor-Hom adjunction; the smooth and proper base change formulas.
(3)    Constructible sheaves on complex algebraic varieties.
(4)    Verdier duality.
(5)    Nearby and vanishing cycles; unipotent and tame variants.
(6)    Compatibility of vanishing cycles with the six functors and Verdier duality.
(7)    Thom-Sebastiani for vanishing cycles.
(8)    Hyperbolic localization for vanishing cycles.

2. Prerequisites

•   Singular co/homology; basic homological algebra (chain complexes, co/homology, Tor/Ext).

•   Basics about complex manifolds or complex algebraic varieties.

3. Grading Scheme

Grades will be determined based on exercise sheets.

4. Course Goal

The goal is to develop a working knowledge of constructible sheaves and vanishing cycles sheaves.  The student will be able to starting reading papers where the theory is applied in enumerative geometry, geometric representation theory, or other areas.

5. Reference material (textbooks)
·       Kashiwara, Schapira, Sheaves on manifolds.
·       Achar, Perverse sheaves and applications to representation theory.

6. Registration

https://forms.gle/YrPFLZvq9knUMG1M6

7. Credit: 3

Course No.: NCTS5058

Course ID: V41 U2060