Complex Analysis and Riemann Surfaces: A Graduate Path to Algebraic Geometry

TL;DR

This paper provides a compute-first, explicit approach to understanding Riemann surfaces and their role in algebraic geometry, bridging complex analysis with geometric and algebraic structures for graduate students.

Contribution

It offers a detailed, computation-driven pathway from classical complex analysis to advanced algebraic geometry, emphasizing explicit constructions and examples.

Findings

Explicit construction of Riemann surfaces via branched coverings

Detailed proof of the Riemann-Roch theorem with applications

Development of the Jacobian, Abel-Jacobi theory, and theta functions

Abstract

These lecture notes present a computation driven pathway from classical complex analysis to the theory of compact Riemann surfaces and their connections to algebraic geometry. The exposition follows a compute first then abstract philosophy, in which analytic and geometric structures are introduced through explicit calculations and local models before being organized into conceptual frameworks. The notes begin with the foundations of complex analysis, including holomorphic functions, Cauchy theory, power series, residues, and contour integration, with an emphasis on hands on techniques such as Laurent expansions, residue calculus, and branch cut methods. These analytic tools are then used to construct Riemann surfaces explicitly via branched coverings and gluing constructions, which serve as recurring test cases throughout the text. Differential forms, Stokes theorem, curvature, and the Gauss Bonnet theorem provide the geometric bridge to Hodge theory, culminating in a detailed and self contained treatment of the Hodge Weyl theorem on compact Riemann surfaces, including weak formulations, regularity, and concrete examples. The algebraic geometric core develops holomorphic line bundles, divisors, the Picard group, and sheaves, followed by Cech and sheaf cohomology, the exponential sequence, and de Rham and Dolbeault theories, all treated with explicit computations. The Riemann Roch theorem is presented with full proofs and applications, leading to the construction of the Jacobian, Abel Jacobi theory, theta functions, and the correspondence between Riemann surfaces, algebraic curves, and Galois coverings. Originating from collaborative study groups associated with the Enjoying Math community, these notes are intended for graduate students seeking a concrete and unified route from complex analysis to algebraic geometry.