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.

• Homework 1, due 2/8.
• Homework 2, due 2/22.
• Homework 3, due 3/8.
• Homework 4, due 3/31.
• Homework 5, due 4/19.
• Homework 6, not assigned (for practice only)

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)

• 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)