Integrable systems and complex geometry

TL;DR

This paper explores the deep connections between complex geometry and integrable systems, introducing new examples, algebraic methods, and the concept of algebraic complete integrability in Hamiltonian systems.

Contribution

It presents new integrable systems based on Lax representations, applies Lie algebra methods to analyze integrability, and discusses the generalized notion of algebraic complete integrability.

Findings

New integrable systems via spectral curves and Lax pairs

Criteria for algebraic complete integrability of Hamiltonian systems

Extension to coverings of algebraic integrability systems

Abstract

In this paper, we discuss an interaction between complex geometry and integrable systems. Section 1 reviews the classical results on integrable systems. New examples of integrable systems, which have been discovered, are based on the Lax representation of the equations of motion. These systems can be realized as straight line motions on a Jacobi variety of a so-called spectral curve. In section 2, we study a Lie algebra theoretical method leading to integrable systems and we apply the method to several problems. In section 3, we discuss the concept of the algebraic complete integrability (a.c.i.) of hamiltonian systems. Algebraic integrability means that the system is completely integrable in the sens of the phase space being folited by tori, which in addition are real parts of a complex algebraic tori (abelian varieties). The method is devoted to illustrate how to decide about the a.c.i. of hamiltonian systems and is applied to some examples. Finally, in section 4 we study an a.c.i. in the generalized sense which appears as covering of a.c.i. system. The manifold invariant by the complex flow is covering of abelian variety.