Introduction to Riemann Surfaces - 2023-2024

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 11/09 : Introduction, Field of meromorphic functions.

cours 15/09 : Elliptic functions I.

cours 18/09 : Elliptic functions II.

cours 22/09 : Topology of surfaces.

cours 25/09 : Riemann-Hurwitz formula, Riemann surfaces as branched covers, the field of meromorphic functions of a branched cover.

cours 29/09 : Statement of the uniformization theorem. Plane (affine and projective) algebraic curves.

cours 02/10 : Bezout's theorem, Plucker's formula.

cours 06/10 : Differential, holomorphic and meromorphic forms. De Rham cohomology on a surface. Residue of a meromorphic form.

cours 09/10 : Poisson equation. Hodge decomposition. Existence of meromorphic functions.

cours 13/10 : Riemann-Roch theorem, applications, canonical embedding.

cours 16/10 : Line bundles and divisors. Sheaves (will not be a subject in the exam).

cours 20/10 : Cohomology of sheaves and Riemann-Roch (will not be a subject in the exam).

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