Integrable Systems in the Realm of Algebraic Geometry

This book treats the general theory of Poisson structures and integrable systems on affine varieties in a systematic way. Special attention is drawn to algebraic completely integrable systems. Several integrable systems are constructed and studied in detail and a few applications of integrable systems to algebraic geometry are worked out. In the second edition, some of the concepts in Poisson geometry are clarified by introducing Poisson cohomology; the Mumford systems are constructed from the algebra of pseudo-differential operators, which clarifies their origin; a new explanation of the multi Hamiltonian structure of the Mumford systems is given by using the loop algebra of sl(2); and finally, the Goedesic flow on SO(4) is added to illustrate the linearization algorithm and to give another application of integrable systems to algebraic geometry.