1. Complex manifolds. Basic definitions and properties. Examples. Holomorphic vector bundles.
2. Sheaf cohomology. Basic homological algebra. De Rham cohomology. Presheaves and sheaves. Cech cohomology. Cup product. Chern classes of vector bundles.
3. Divisors. Divisors, meromorphic functions and line bundles on Riemann surfaces. The higher dimensional case. Bertini's theorem. The adjunction formula.
4. Algebraic curves. Serre duality. The Kodaira embedding. Branched coverings. The Riemann-Hurwitz formula. The g=0 and g=1 cases. The Weierstrass representation of elliptic curves. Jacobian varieties. Blow-up. Nodal curves.
5. Algebraically integrable systems. Spectral curves. Flow linearization on the Jacobian variety. Examples.