arXiv2026
This book develops a computation-driven route from classical complex analysis to compact Riemann surfaces and algebraic geometry. It begins with holomorphic and meromorphic functions, Cauchy theory, power and Laurent series, residues, contour integration, branch cuts, harmonic functions, conformal mapping, invariant metrics, the Riemann mapping theorem, elliptic functions, and analytic continuation. Differential forms and Stokes' theorem lead to de Rham cohomology, curvature, Gauss-Bonnet, and Hodge theory. Coordinate calculations relate first fundamental forms to Hermitian and Kahler metrics and reduce curvature under conformal change to Poisson and Liouville equations; the sphere and torus are worked out explicitly. The Hodge-Weyl theorem is developed through weak formulations, elliptic regularity, harmonic representatives, top-degree cohomology, and the one-dimensional Calabi-Yau equation. The cohomological chapters introduce complex manifolds, line bundles, divisors, the Picard group, sheaves, Cech cohomology, derived functors, fine resolutions, de Rham and Dolbeault cohomology, the exponential sequence, canonical bundles, Chern classes, Stein Riemann surfaces, and the Mittag-Leffler problem. Branched covers and gluing remain recurring models. Duality and several proofs of Riemann-Roch lead to surface Riemann-Roch and intersection theory. The final parts treat Jacobians, Abel-Jacobi theory, algebraic curves, local intersection multiplicity, theta and Weierstrass functions, Mellin transforms and the zeta functional equation, Galois coverings, modular curves, and cohomological representations. Definitions and global theorems are paired with calculations, examples, and exercises for advanced undergraduates and beginning graduate students.