On the Local Geometry of a Bihamiltonian Structure

  • Israel M. Gelfand
  • Ilya Zakharevich

Abstract

We give several examples of bihamiltonian manifolds and show that under very mild assumptions a bihamiltonian structure in “general position” is locally of one of these types. This shows, in particular, that a bihamiltonian manifold in general position is always a moduli space of some kind. In the even-dimensional case it is a Hubert scheme of a surface, in the odd-dimensional case it is a sub- cotangent bundle of a moduli space of rational curves on a surface.

Keywords

Filtration Manifold Soliton Intersection Line Allo 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I. M. Gelfand and I. S. Zakharevich, Spectral theory of a pencil of third-order skew-symmetric differential operators on s 1, Functional Analysis and its Applications 23 (1989), no. 2, 85–93.MathSciNetCrossRefGoogle Scholar
  2. 2.
    I. M. Gelfand and I. S. Zakharevich —, Webs, Veronese curves, and bi-Hamiltonian systems, Journal of Functional Analysis 99 (1991), no. 1, 150–178.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Israel M. Gelfand and Ilya Zakharevich, The spectral theory for a pencil of skew-symmetrical differential operators of third order, preprint MSRI-06627-91, MSRI, Berkeley, CA, 94720, 1991.Google Scholar
  4. 4.
    Alexander Goncharov, private communication, 1990.Google Scholar
  5. 5.
    Kunihiko Kodaira, Complex manifolds and deformation of complex structures, Grundlehren der mathematischen Wissenschaften, vol. 283, Springer-Verlag, New York, 1986.Google Scholar
  6. 6.
    F. Magri, Geometry and soliton equations, La Mécanique Analytique de Lagrange et son héritage, Collège de France, September 1988.Google Scholar
  7. 7.
    Henri McKean, private communication, 1990.Google Scholar
  8. 8.
    Francisco-Javier Turiel, Classification locale d’un couple de formes symplectiques Poisson-compatibles, Comptes Rendus des Seances de l’Academie des Sciences. Serie I. Mathematique 308 (1989), no. 20, 575–578.MathSciNetMATHGoogle Scholar
  9. 9.
    Alan Weinstein, The local structure of Poisson manifolds, Journal of Differential Geometry 18 (1983), no. 3, 523–557.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Israel M. Gelfand
    • 1
  • Ilya Zakharevich
    • 2
  1. 1.Dept. of MathematicsRutgers University, Hill CenterNew BrunswickUSA
  2. 2.Dept. of MathematicsMITCambridgeUSA

Personalised recommendations