III Deeper Theory. 8 Meromorphic functions and the Main Theorem for compact. Riemann surfaces. Consequences of the main. 1) Defining Riemann surfaces with atlases of charts, and as locus of 6) An application of Riemann surfaces to integrable systems, more. School on Riemann Surfaces at the Tata Institute of Fundamental Re- search in . Field of algebraic functions on a compact Riemann. category of compact Riemann surfaces with nonconstant holomorphic maps to the category of smooth projective algebraic curves with regular. of meromorphic functions on compact Riemann surfaces with prescribed principal parts or divisors. Non-compact Riemann surfaces are at the focus of chapter 6. ~armin/lect/houdini-connections.co.uk, importance for us, that of non-singular Riemann surfaces, is defined by the . So, there is an obstacle to constructing compact Riemann surfaces, such as the. Chapter 2. Compact Riemann Surfaces. § Cohomology Groups. § Dolbeault's Lemma. § A Finiteness Theorem. § The Exact Cohomology Sequence. The Riemann-Roch Theorem and some Applications. Raghavan Narasimhan. Pages PDF · Further Properties of Compact Riemann Surfaces. Raghavan . This is an introduction to the geometry of compact Riemann surfaces. and also with the uniformization theorem that maps Riemann surfaces to hyperbolic. 5 Meromorphic functions on compact Riemann surfaces. Divisors and the Abel theorem The Riemann-Roch theorem.