Integrable systems in the realm of algebraic geometry

Pol Vanhaecke

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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[目次]

  • 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)
  • ...

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この本の情報

書名 Integrable systems in the realm of algebraic geometry
著作者等 Vanhaecke, Pol
シリーズ名 Lecture notes in mathematics
出版元 Springer
刊行年月 c2001
版表示 2nd ed
ページ数 x, 256 p.
大きさ 24 cm
ISBN 3540423370
NCID BA53179448
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言語 英語
出版国 ドイツ
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