Math 241 - Complex Manifolds - Spring 2011
D. Auroux -
Tue. & Thu., 11-12:30pm, Room 81 Evans
Instructor:
Denis Auroux (auroux@math.berkeley.edu)
Office: 817 Evans.
Office hours: by appointment.
Course outline
The course will begin with Riemann surfaces, then proceed with
higher-dimensional complex manifolds. The topics include: differential
forms, Cech and Dolbeault cohomology, divisors and line bundles,
Riemann-Roch, vector bundles, connections and curvature, Kahler-Hodge
theory, Lefschetz theorems, Kodaira theorems.
Prerequisites: Math 214 (Differentiable Manifolds) and 215A (Algebraic
Topology)
Grading: based on homework
Recommended texts
- D. Huybrechts, Complex geometry: an introduction, Springer
Universitext.
Note: the university library has this text available as an e-book
here (from campus only);
you can order a deeply discounted soft-bound printed copy from Springer
from the e-book download page.
- O. Forster, Lectures on Riemann surfaces, Springer GTM.
Homework
Homework will be due every other week. Assignments
will be posted here.
Approximate schedule
Part 1: Riemann surfaces
- 1/18: About the course; Riemann surfaces, elementary
properties of holomorphic mappings (Forster §1-2)
- 1/20: Branched and unbranched coverings of Riemann
surfaces (Forster §3-4)
- 2/1: Sheaves; analytic continuation
(Forster §6-7)
- 2/3: Differential forms on Riemann surfaces
(Forster §9-10)
- 2/8: Sheaf cohomology; Dolbeault's lemma
(Forster §12-13)
- 2/10: Exact sequences and sheaf cohomology
(Forster §15)
- 2/15: Dolbeault's theorem; holomorphic vector bundles
(Forster §29)
- 2/17: Divisors and line bundles; Riemann-Roch theorem
(Forster §16)
- 2/22: Serre duality
(Forster §17)
- 3/1: Abel's theorem
(Forster §20)
- 3/3: The Jacobi inversion problem
(Forster §21)
Part 2: Complex manifolds
- 3/8: Holomorphic functions of several variables: some
properties (Huybrechts §1.1)
- 3/10: Complex and Hermitian vector spaces, types and
decomposition (Huybrechts §1.2)
- 3/15: Differential forms on almost-complex manifolds
(Huybrechts §1.3)
- 3/17: Complex manifolds: definition and examples
(Huybrechts §2.1)
- 3/29: Holomorphic vector bundles
(Huybrechts §2.2)
- 3/31: Projective space; blowups
(Huybrechts §2.4-2.5)
- 4/5: Divisors and line bundles
(Huybrechts §2.3)
- 4/7: Differential calculus on complex manifolds
(Huybrechts §2.6)
- 4/12: Kähler identities
(Huybrechts §3.1)
- 4/14: Hodge theory on Kähler manifolds
(Huybrechts §3.2)
- 4/19: Lefschetz theorems
(Huybrechts §3.3)
- 4/21: Hermitian vector bundles; connections
(Huybrechts §4.1-4.2)
- 4/26: Curvature
(Huybrechts §4.3)
- 4/28: Chern classes
(Huybrechts §4.4)
- 5/3: (optional) Kodaira vanishing and embedding theorems
(Huybrechts §5.2-5.3)
- 5/5 (optional): Deformations of complex structures
(Huybrechts §6.1) (beginning only; Teichmuller space)