Introduction to Riemann Surfaces - 2025-2026

References :

R. Miranda: Algebraic Curves and Riemann Surfaces. Graduate Studies in Mathematics (1995).

S. Donaldson: Riemann surfaces. Oxford Graduate Texts in Mathematics (2011).

O. Forster: Lectures on Riemann surfaces. Graduate texts in mathematics. Springer (1981).

Un poly par N. Bergeron et A. Guilloux.


Le polycopié sera actualisé chaque semaine.

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cours 09/09 : Introduction, Field of meromorphic functions.

cours 11/09 : Divisors, Elliptic functions.

cours 18/09 : Abel's theorem, Riemann-Roch theorem for an elliptic curve.

cours 23/09 : Topology of surfaces, Riemann-Hurwitz formula.

cours 26/09 : Riemann surfaces as branched covers, the field of meromorphic functions of a branched cover.

cours 30/09 : Statement of the uniformization theorem. Affine and projective algebraic curves.

cours 02/10 : Any smooth projective algebraic curve has an embedding into CP^3. Intersection divisor, Bezout's theorem, Plucker's formula.

cours 07/10 : Holomorphic vector bundles. Differential, holomorphic and meromorphic forms. Residue of a meromorphic form.

cours 09/10 : Riemann-Roch theorem, applications, canonical embedding. Holomorphic line bundles and divisors.

cours 14/10 : De Rham cohomology on a surface. Poisson equation. Hodge decomposition.

cours 16/10 : Existence of meromorphic functions. Existence of meromorphic differentials. Periods, Riemann bilinear relations. Proof of Riemann-Roch theorem.

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--> Exams 2023, 2024.