# curso "Abelian varieties in algebraic integrable systems"

Contents: 6 lectures, each one 55 min., given in Spanish with slides in English.

*Description of the first 3 lectures*.

Lecture 1. Algebraic curves, Abelian varieties, and algebraic integrable systems.

- A first example: The classical Euler top on so(3). Equations of motion, elliptic curve, linearization.

- Abelian varieties: definitions, basic properties, examples. Jacobi varieties (algebraic and analytical description).

- Algebraic integrable systems. Examples. Historical remarks.

- Exercises.

Lecture 2. Algebraic integrable systems and the Painlevé property.

- More properties of Abelian varieties: divisors, linear systems, embedding to projective space.

- The essence of the Kovalevskaya--Painlevé (KP) analysis: Laurent series solutions, Kowalewski exponents, the theorems of

Adler & van Moerbeke and of Yoshida, and their generalizations.

- Geometric interpretation of the KP analysis.

- Examples: the Euler top again, the Clebsch integrable case of the Kirchoff equations.

Lecture 3. Lax representation and spectral curves.

- What really is a Lax representation of an integrable system. Examples.

- The eigenvector map to the Jacobian of the spectral curve S. Linearization of the flow on the Jacobian.

- The problem of separation of variables.

- Spectral curves with an involution and Prym subvarieties of the Jacobian.

- Relation between the Jacobian of S and the complex invariant manifold of the system.