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

TL;DR

The notes trace a compute‑first pathway from classical complex analysis to the theory of compact Riemann surfaces and their links to algebraic geometry, illustrating how explicit calculations (Laurent expansions, residues, branch cuts) illuminate global invariants. A central theme is translating analytic data into geometric structures via differential forms, Stokes’ theorem, and cohomology, culminating in Riemann–Roch, the Jacobian, Abel–Jacobi theory, and theta functions. The material emphasizes hands‑on, test‑case reasoning (e.g., two‑sheeted covers, genus computations) to make abstract concepts like sheaves and cohomology tangible. Originating from collaborative study groups within Enjoying Math, the notes aim to provide a concrete, unified route from complex analysis to algebraic geometry, balancing calculations with conceptual frameworks to reveal the deep unity of the subject.

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.

Figures (7)

• Figure 1: A $2$-form is an alternating bilinear map: it measures signed oriented area.
• Figure 2: $dx\wedge dy$ measures signed area; swapping $(v,w)$ changes the sign.
• Figure 3: The closed $1$-form $\eta$ detects topology: loops winding around the origin give nonzero integral.
• Figure 4: The complex differentials $dz,d\bar{z}$ as linear combinations of $dx,dy$.
• Figure 5: The rotational field corresponds to $\eta=(x\,dy-y\,dx)/(x^2+y^2)$ and its circulation is detected by $\oint dz/z$.

Theorems & Definitions (957)

• Definition 1.1: Domain
• Definition 1.2: Complex differentiability and holomorphicity
• Lemma 1.4: Real-linear maps that are complex multiplications
• proof
• Definition 1.6: Wirtinger derivatives
• Lemma 1.7: CR equations in Wirtinger form
• proof
• Example 1.8: A holomorphic and a non-holomorphic map
• Theorem 1.10: CR equations imply complex differentiability
• proof