Géométrie et intégrabilité algébrique

Abstract

In this paper, I present an overview of the active area of interactions between algebraic geometry and algebraic completely integrable systems. These are integrable systems whose trajectories are straight line motions on complex algebraic tori (abelian varieties). We make, via the Kowalewski-Painlevé analysis, a detailed study of the level manifolds of the system. These manifolds are described explicitly as being affine part of complex algebraic tori and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-van Moerbeke method’s which will be used is primarily analytical but heavily inspired by algebraic geometrical methods. We will also discuss several interesting examples of algebraic completely integrable systems.

Keywords

• integrable systems
• curves
• abelian varieties
• jacobian varieties
• prym varieties
• embeddings
• kähler manifolds
• rigid body