Note for Riemannian Surface
The note covers the following:
Part I:
1. Basic complex analysis, harmonic functions, Riemann mapping theorem
2. Definitions of Riemann surface, meromorphic function/differential, subharmonic to harmonic(Perron method), uniformization theorem
3. Divisor, Hodge decomposition theorem, Riemann-Roch formula and proof
4.Applications of Riemann-Roch: various merofunctions on Riemann surface, function field, ellptic function, embedding, Riemann-Hurwitz formula, bi-linear relations, Abel-Jacobi theorem
Part II:
1. Definitions of bundle, construct line bundle from divisor, sheaf and presheaf, Cech cohomology, Doubealt theorem, Euler number
2. Hermitian metric, connection and curvature of line bundle, duality and vanishing.