Lectures on compact Riemann surfaces

B. Eynard (1)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681

This is an introduction to the geometry of compact Riemann surfaces. We largely follow the books [8, 9, 10]. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations. 2) Space of meromorphic functions and forms, we classify them with the Newton polygon. 3) Abel map, the Jacobian and Theta functions. 4) The Riemann-Roch theorem that computes the dimension of spaces of functions and forms with given orders of poles and zeros. 5) The moduli space of Riemann surfaces, with its combinatorial representation as Strebel graphs, and also with the uniformization theorem that maps Riemann surfaces to hyperbolic surfaces. 6) An application of Riemann surfaces to integrable systems, more precisely finding sections of an eigenvector bundle over a Riemann surface, which is known as the "algebraic reconstruction" method in integrable systems, and we mention how it is related to Baker-Akhiezer functions and Tau functions.

Domaines: Physique mathématique [math-ph]

B. Eynard. Lectures on compact Riemann surfaces. Doctoral. Paris-Saclay, France. 2018. ⟨cel-02013559⟩