Note for Riemannian Surface

The note covers the following:   
   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
   5. Definitions of bundle, construct line bundle from divisor, sheaf and presheaf, Cech cohomology, Doubealt theorem, Euler number
   6. Hermitian metric, connection and curvature of line bundle, duality and vanishing.