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. TOC:Introduction.- Integrable Hamiltonian systems on affine Poisson varietie: Affine Poisson varieties and their morphisms; Integrable Hamiltonian systems and their morphisms; Integrable Hamiltonian systems on other spaces.- Integrable Hamiltonian systems and symmetric products of curves: The systems and their integrability; The geometry of the level manifolds .- Interludium: the geometry of Abelian varieties: Divisors and line bundles; Abelian varieties; Jacobi varieties; Abelian surfaces of type (1,4).- Algebraic completely integrable Hamiltonian systems: A.c.i. systems; Painlev analysis for a.c.i. systems; The linearization of two-dimensional a.c.i. systems; Lax equations.- The Mumford systems: Genesis; Multi-Hamiltonian structure and symmetries; The odd and the even Mumford systems; The general case .- Two-dimensional a.c.i. systems and applications: The genus two Mumford systems; Application: generalized Kummersurfaces; The Garnier potential; An integrable geodesic flow on SO(4);...
Inhalt
I. Introduction: II. Integrable Hamiltonian systems on affine Poisson varieties: 1. Introduction.- 2. Affine Poisson varieties and their morphisms.- 3. Integrable Hamiltonian systems and their morphisms.- 4. Integrable Hamiltonian systems on other spaces.- III. Integrable Hamiltonian systems and symmetric products of curves: 1. Introduction.- 2. The systems and their integrability.- 3. The geometry of the level manifolds.- IV. Interludium: the geometry of Abelian varieties 1. Introduction.- 2. Divisors and line bundles.- 3. Abelian varieties.- 4. Jacobi varieties.- 5. Abelian surfaces of type (1,4).- V. Algebraic completely integrable Hamiltonian systems: 1. Introduction.- 2. A.c.i. systems.- 3. Painlev analysis for a.c.i. systems.- 4. The linearization of two-dimensional a.c.i. systems.- 5. Lax equations.- VI. The Mumford systems 1. Introduction.- 2. Genesis.- 3. Multi-Hamiltonian structure and symmetries.- 4. The odd and the even Mumford systems.- 5. The general case.- VII. Two-dimensional a.c.i. systems and applications 1. Introduction.- 2. The genus two Mumford systems.- 3. Application: generalized Kummersurfaces.- 4. The Garnier potential.- 5. An integrable geodesic flow on SO(4).- 6. The Hnon-Heiles hierarchy.- 7. The Toda lattice.- References.- Index.