This course is taken in sequence, part 1 in the fall, and part 2 in the spring.

COMPLEX ANALYSIS AND RIEMANN SURFACES I

Local theory of holomorphic functions
The Riemann surface of $w^2=z(z-1)(z-\lambda)$
Construction of holomorphic forms and meromorphic forms with prescribed poles
Abel’s theorem
The Jacobi inversion theorem

Function theory on tori:

• The point of view of Weierstrass
• The point of view of Jacobi: $\theta$ functions
• The point of view of partial differential equations

Line bundles, connections, metrics, curvature, first Chern class
The Riemann-Roch theorem, applications
The Riemann-Roch theorem as an index theorem
A heat equation proof of the Riemann-Roch theorem
Function theory on Riemann Surfaces of higher genus:
Theta functions, the theta divisor, the prime form
Theta characteristics and spinors

COMPLEX ANALYSIS AND RIEMANN SURFACES II

Holomorphic vector bundles on complex manifolds
Connections and curvature, characteristic classes and Chern-Weil theory
The Uniformization theorem
The Ricci curvature and the Calabi conjecture:
Statement and complete proof; estimates for the complex Monge-Ampere equation
Hermitian-Einstein metrics and Mumford stability
The Donaldson-Uhlenbeck-Yau theorem
The proof by Uhlenbeck and Yau

Depending on the year, the last part of II (Hermitian-Einstein metrics, etc) may be replaced by a treatment of the $\bar\partial$ equation, as follows:

• Bochner-Kodaira formulas
• H\”ormander’s estimates for the $\bar\partial$ equation
• Singular metrics and multiplier ideal sheaves
• The Kodaira imbedding theorem
• The Ohsawa-Takegoshi extension theorem

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