Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T17:38:34.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Jacobians of singular spectral curves and completely integrable systems

Published online by Cambridge University Press:  20 January 2009

Olivier Vivolo
Olivier Vivolo
Affiliation:
Laboratoire Emile Picard, URA CNRS 5580, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex, France (vivolo@picard.ups-tlse.fr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Consider an isospectral manifold formed by matrices M ∈ glr(ℂ)[x] with a fixed leading term. The description of such a manifold is well known in the case of a diagonal leading term with different eigenvalues. On the other hand, there are many important systems where this term has multiple eigenvalues. One approach is to impose conditions in the sub-leading term. The result is that the isospectral set is a smooth manifold, bi-holomorphic to a Zariski open subset of the generalized Jacobian of a singular curve.

Keywords

integrabilityJacobian variety
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 3 , October 2000 , pp. 605 - 623
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1.Adams, M. R., Harnad, J. and Hurtubise, J., Isospectral Hamiltonian flows in finite and infinite dimension, Commun. Math. Phys. 134 (1990), 555–585.CrossRefGoogle Scholar
2.Adler, M. and van Moerbeke, P., Linearization of Hamiltonian systems, Jacobi varieties and representation theory, Adv. Math. 38 (1980), 318–379.CrossRefGoogle Scholar
3.Beauville, A., Jacobiennes des courbes spectrales et systèmes Hamiltoniens complétement intégrales, Acta. Math. 164 (1990), 211–235.CrossRefGoogle Scholar
4.Belokolos, E. D., Bobenko, A. I., Enosl'skiĭ, V. Z., Its, A. R. and Matveev, V. B., Algebro-geometric approach to nonlinear integrable equations (Springer, 1994).Google Scholar
5.Bobenko, A. I., Reyman, A. G., Semenov, M. A. and Shansky, T., The Kowalewski top 99 years later: a Lax pair, generalzations and explicit solutions, Commun. Math. Phys. 122 (1989), 312–354.CrossRefGoogle Scholar
6.Bottacin, F., Symplectic geometry on moduli spaces of stable pairs, Ann. Sci. Ecole Norm. Sup. 28 (1995), 391–433.CrossRefGoogle Scholar
7.Donagi, R. and Markman, E., Spectral covers, algebraically completely integrable, Hamiltonian systems and moduli of bundles, Lectures Notes in Mathematics, vol. 1620 (Springer, 1993).Google Scholar
8.Gavrilov, L., Generalized Jacobians of spectral curves and completely integrable systems, Math. Z. 230 (1999), 487–508.CrossRefGoogle Scholar
9.Hitchin, N., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91–114.CrossRefGoogle Scholar
10.Jacobi, C., Vorlesungen über Dynamik (G. Reimer, Berlin, 1891). (Réimpression: Chelsea, New York, 1967.)Google Scholar
11.Lang, S., Unramified class field theory over function fields in several variables. Ann. Math. 64 (1956), 285–325.CrossRefGoogle Scholar
12.Lang, S., Sur les séries L d'une variété algébrique, Bull. Soc. Math. France 84 (1956), 385–407.CrossRefGoogle Scholar
13.Markman, E., Spectral curves and integrable systems, Compositio Math. 93 (1994), 255–290.Google Scholar
14.Reyman, A. G., Semenov, M. A. and Shansky, T., Group theoretical methods in the theory of finite dimensional integrable systems, in Dynamical systems, vol. VII, Encyclopedia of Mathematical Sciences, vol. 16 (Springer, 1994).Google Scholar
15.Rosenlicht, M., Generalized Jacobian varieties, Ann. Math. 59 (1954), 505–530.CrossRefGoogle Scholar
16.Rosenlicht, M., A universal mapping property of generalized Jacobian varieties, Ann. Maths 66 (1957), 80–88.CrossRefGoogle Scholar
17.Serre, J. P., Groupes algébriques et corps de classes (Hermann, Paris, 1959).Google Scholar
18.van Moerbeke, P. and Mumford, D., The spectrum of difference operators and algebraic curves, Acta Math. 143 (1979), 93–154.CrossRefGoogle Scholar
19.Vivolo, O., Systèmes intégrables et courbes algébriques, PhD thesis, Université de Toulouse III (1997).Google Scholar
20.Weil, A., Variétés abéliennes et courbes algébrique (Hermann, Paris, 1948).Google Scholar