Lecture 3 (Thu, Mar 6, 2008)

Global theory of linear systems: holomorphic vector bundles

1. Definitions. Gluing bundles from cylindrical charts.
2. Matrix cocycles and their equivalence.
3. Operations on bundles vs. operations with cocycles.
4. Example: linear bundles over . Degree.
5. Sections (holomorphic and meromorphic) of holomorphic bundles.
6. Triviality of holomorphic vector bundles over and classification of bundles over : Cartan and Birkhoff–Grothendieck theorems.

Recommended reading: the subject is treated in various sources with accent on analytic, geometric or algebraic side of it. You can choose your favorite textbook or one of the following expositions.

1. O. Forster, Riemann surfaces, §§29-30 (analytic treatment).
2. P. Griffiths & M. Harris, Principles of Algebraic Geometry, §0.5 (algebraic “neoclassical”).
3. R. O. Wells, Differrential Analysis on Complex Manifolds, §2.
4. A. Hatcher, Vector bundles and -theory (rather advanced topological exposition).
5. Parts 17A-17F and 17J from Section 17 from the Book (printing disabled).