Math 545 -- Complex Geometry

Fall, 2014

This course will present an introduction to the theory of complex manifolds. It is aimed at students interested in complex or algebraic geometry. Throughout the class, we will emphasize concrete examples and applications of the general theory. A companion course, taught by Radu Laza in the Spring, will focus on the more algebraic side of the theory.

The tentative syllabus is as follows:

1. Holomorphic functions of several complex variables.

2. Definition and examples of complex manifolds.

3. Holomorphic line bundles and vector bundles. Embeddings into projective space.

4. Complex and Hermitian linear algebra. Differential forms on a complex manifold. Dolbeaut cohomology.

5. Kahler manifolds, Kahler identities.

6. The Hodge theorem.

7. Applications: the Kodaira vanishing and Kodaira embedding theorems.

In a general way, we will follow the approach of Griffiths-Harris and Huybrechts, but we will circulate notes for the class. These are available from the link below.

This course will present an introduction to the theory of complex manifolds. It is aimed at students interested in complex or algebraic geometry. Throughout the class, we will emphasize concrete examples and applications of the general theory. A companion course, taught by Radu Laza in the Spring, will focus on the more algebraic side of the theory.

The tentative syllabus is as follows:

1. Holomorphic functions of several complex variables.

2. Definition and examples of complex manifolds.

3. Holomorphic line bundles and vector bundles. Embeddings into projective space.

4. Complex and Hermitian linear algebra. Differential forms on a complex manifold. Dolbeaut cohomology.

5. Kahler manifolds, Kahler identities.

6. The Hodge theorem.

7. Applications: the Kodaira vanishing and Kodaira embedding theorems.

In a general way, we will follow the approach of Griffiths-Harris and Huybrechts, but we will circulate notes for the class. These are available from the link below.