Compact riemann surfaces

Abstract

In this paper, we study compact Riemann surfaces, namely a compact connected complex manifold of dimension 1. We first introduce the notion of line bundles and divisors on compact Riemann surfaces, and observe connections with sheaf cohomologies. Next, we prove the Riemann-Roch theorem which plays a significant role in complex analysis and algebraic geometry. We then examine some important applications and consequences of the Riemann-Roch theorem such as Riemann-Hurwitz formula or Clifford`s theorem. In the last few chapters, we introduce the Jacobian of a compact Riemann surface and the Abel-Jacobi map which connects a compact Riemann surface and its Jacobian. We examine some properties of the Abel-Jacobi map and related topic; the theta divisor.