Vector Bundles on Riemann Surfaces

1. Differentiable Manifolds 2 2. Complex Manifolds 3 2.1. Riemann Surfaces of Genus One 4 2.2. Constructing Riemann Surfaces as Curves in P 6 2.3. Constructing Riemann Surfaces as Covers 9 2.4. Constructing Riemann Surfaces by Glueing 10 3. Topological Vector Bundles 11 3.1. The Tangent and Cotangent Bundles 13 3.2. Interlude: Categories, Complexes and Exact Sequences 14 3.3. Metrics on Vector Bundles 15 3.4. The Degree of a Line Bundle 16 3.5. The Determinantal Line Bundle 17 3.6. Classification of Topological Vector Bundles on Riemann Surfaces 18 3.7. Holomorphic Vector Bundles 19 3.8. Sections of Holomorphic Vector Bundles 20 4. Sheaves 21 4.1. Cech Cohomology 23 4.2. Line Bundles and Cech Cohomology 27 4.3. Riemann-Roch and Serre Duality 29 4.4. Vector bundles, locally free sheaves and divisors 30 4.5. A proof of Riemann-Roch for curves 33 5. Classifying vector bundles on Riemann surfaces 34 5.1. Grothendieck’s classification of vector bundles on P 34 5.2. Atiyah’s classification of vector bundles on elliptic curves 35 References 36