Integrable Systems in the realm of Algebraic Geometry

Pol Vanhaecke (auth.)3540423370, 9783540423379

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 introducting 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 Goedesic flow on SO(4) is added to illustrate the linearizatin algorith and to give another application of integrable systems to algebraic geometry.

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Table of contents :
Front Matter….Pages N2-viii
Introduction….Pages 1-15
Integrable Hamiltonian systems on affine Poisson varieties….Pages 17-65
Integrable Hamiltonian systems and symmetric products of curves….Pages 67-93
Interludium: the geometry of Abelian varieties….Pages 95-122
Algebraic completely integrable Hamiltonian systems….Pages 123-138
The master systems….Pages 139-169
The Garnier and Hénon-Heiles potentials and the Toda lattice….Pages 171-208
Back Matter….Pages 209-221