Integrable systems and complex geometry

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.