Introduction to Riemann Surfaces - 2024-2025

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 16/09 : Divisors, Elliptic functions.

cours 19/09 : Elliptic functions, Abel's theorem.

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

cours 27/09 : Riemann surfaces as branched covers, the field of meromorphic functions of a branched cover.

cours 30/09 : Statement of the uniformization theorem. Plane (affine and projective) algebraic curves.

cours 04/10 : Bezout's theorem, Plucker's formula. Holomorphic vector bundles.

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

cours 11/10 : De Rham cohomology on a surface. Poisson equation. Hodge decomposition. Existence of meromorphic functions.

cours 14/10 : Existence of meromorphic differentials. Riemann-Roch theorem, applications, canonical embedding.

cours 18/10 : Periods, Riemann bilinear relations. Proof of Riemann-Roch theorem.

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Exam 2023.