zbMATH — the first resource for mathematics

Prym varieties and applications. (English) Zbl 1149.14045
The author surveys the theory of the Prym varieties of ramified double coverings of algebraic curves (another source of information on these varieties is the classical book “Theta functions on Riemann surfaces” [J. D. Fay, Lect. Notes Math. 352, Springer-Verlag, Berlin (1973; Zbl 0281.30013)] and their appearance in the theory of integrable systems. Therewith he discusses only finite-dimensional integrable systems, in particular, the Hénon-Heiles system, the Kowalewski rigid body motion and the Kirchhoff equations of motion of a solid body in an ideal fluid and puts aside soliton equations which are integrable in terms of Prym theta functions corresponding to ramified coverings (the most known of them is the Novikov-Veselov equation) and which were recently used by Krichever for solving the Riemann-Schottky type problems for such Prym varieties.

14Q05 Computational aspects of algebraic curves
14H40 Jacobians, Prym varieties
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
Full Text: DOI arXiv
[1] Adler, M.; van Moerbeke, P., The complex geometry of the kowalewski – painlevé analysis, Invent. math., 97, 3-51, (1989) · Zbl 0678.58020
[2] Adler, M.; van Moerbeke, P.; Vanhaecke, P., Algebraic integrability, Painlevé geometry and Lie algebras, () · Zbl 1083.37001
[3] Arbarello, E.; Cornalba, M.; Griffiths, P.A.; Harris, J., Geometry of algebraic curves I, (1994), Springer-Verlag · Zbl 0559.14017
[4] Arnold, V.I., Mathematical methods in classical mechanics, (1978), Springer-Verlag · Zbl 0386.70001
[5] Barth, W., Abelian surfaces with \((1, 2)\)-polarization, Conf. on alg. geom., Sendai, 1985, Adv. stud. pure math., 10, 41-84, (1987)
[6] Beauville, A., Prym varieties and the Schottky problem, Invent. math., 41, 149-196, (1977) · Zbl 0333.14013
[7] Beauville, A.; Narasimhan, M.S.; Ramanan, S., Spectral curves and the generalized theta divisor, J. reine angew. math., 398, 169-179, (1989) · Zbl 0666.14015
[8] Belokolos, E.D.; Bobenko, A.I.; Enol’skii, V.Z.; Its, A.R.; Matveev, V.B., Algebro-geometric approach to nonlinear integrable equations, (1994), Springer-Verlag · Zbl 0809.35001
[9] Griffiths, P.A.; Harris, J., Principles of algebraic geometry, (1978), Wiley-Interscience · Zbl 0408.14001
[10] Haine, L., Geodesic flow on \(S O(4)\) and abelian surfaces, Math. ann., 263, 435-472, (1983) · Zbl 0521.58042
[11] Hitchin, N., Stable bundles and integrable sytems, Duke math. J., 54, 91-114, (1987) · Zbl 0627.14024
[12] Kötter, F., Uber die bewegung eines festen Körpers in einer flüssigkeit I, II, J. reine angew. math., 109, 51-81, (1892), 89-111 · JFM 24.0908.01
[13] Kowalewski, S., Sur le problème de la rotation d’un corps solide autour d’un point fixe, Acta math., 12, 177-232, (1989) · JFM 21.0935.01
[14] Lesfari, A., Abelian surfaces and kowalewski’s top, Ann. scient. école norm. sup., Paris sér. 4, 21, 193-223, (1988) · Zbl 0667.58019
[15] Lesfari, A., Completely integrable systems: jacobi’s heritage, J. geom. phys., 31, 265-286, (1999) · Zbl 0937.37046
[16] Lesfari, A., Le théorème d’arnold – liouville et ses conséquences, Elem. math., 58, 1, 6-20, (2003) · Zbl 1112.37043
[17] Lesfari, A., Le système différentiel de Hénon – heiles et LES variétés Prym, Pacific J. math., 212, 1, 125-132, (2003) · Zbl 1070.37040
[18] Mumford, D., Prym varieties I, (), 325-350
[19] Novikov, S.P., Two-dimensional Schrödinger operator and solitons: three-dimensional integrable systems, (), 226-241
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.