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Integrable systems and algebraic surfaces. (English) Zbl 0857.58024

Duke Math. J. 83, No. 1, 19-50 (1996); erratum ibid. 84, No. 3, 815 (1996).
The paper is devoted to thorough analysis of algebraically integrable Hamiltonian systems (i.e., to complex integrable systems for which the compactified and desingularized joint level sets of the Hamiltonians are Abelian varieties with the linear structures given by the Hamiltonian [cf. P. van Moerbeke, Proc. Symp. Pure Math. 49, Pt. 1, 107-131 (1989; Zbl 0688.70012)]), especially to the particular class of examples determined by some natural systems on the coadjoint orbits of a loop algebra of polynomials in one variable \(\lambda\) with values in a finite-dimensional semisimple Lie algebra [cf., e.g., M. Adler and P. van Moerbeke, Adv. Math. 38, 267-317 (1980; Zbl 0455.58017) and ibid., 318-379 (1980; Zbl 0455.58010)]. The reduced coadjoint orbit has a description as an open set of the union in the family of Jacobians, corresponding to a family of curves in the \((z,\lambda)\)-plane, and its symplectic geometry is related to that of the plane.
The author deals with the following problems: Which integrable systems fit into the loop algebra framework, if not, are there other algebraic surfaces which can be invariantly associated to them, and classification of these surfaces. We cannot refer the deep results here. In rough terms, the vanishing of a certain invariant of the integrable system ensures a surface \(Q\) generalising the \((z,\lambda)\)-plane, in the loop case this \(Q\) is rational, and the classification is made according to whether certain curves \(S_h\) in \(Q\) (determined by the Lagrangian fibration of the integrable system) are hyperelliptic or not. Several examples related to recent work by M. R. Adams, J. Harnad, the author (coadjoint orbits), N.J. Hitchin (moduli spaces of stable \({\mathfrak {gl}}(r)\) pairs), E. K. Sklyanin (unusual brackets), S. Makai (\(K-3\) surfaces), and the genus two case are discussed.
[The erratum concerns Proposition 3.5 of the paper].
Reviewer: J.Chrastina (Brno)

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
14H99 Curves in algebraic geometry
70H05 Hamilton’s equations
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[1] M. R. Adams, J. Harnad, and J. Hurtubise, Isospectral Hamiltonian flows in finite and infinite dimensions II. Integration of flows , Comm. Math. Phys. 134 (1990), no. 3, 555-585. · Zbl 0717.58051 · doi:10.1007/BF02098447
[2] M. R. Adams, J. Harnad, and J. Hurtubise, Darboux coordinates and Liouville-Arnold integration in loop algebras , Comm. Math. Phys. 155 (1993), no. 2, 385-413. · Zbl 0791.58047 · doi:10.1007/BF02097398
[3] M. R. Adams, J. Harnad, and J. Hurtubise, Darboux coordinates and Serre duality , CRM preprint, 1994.
[4] M. R. Adams, J. Harnad, and J. Hurtubise, Coadjoint Orbits, Spectral Curves and Darboux Coordinates , The Geometry of Hamiltonian Systems (Berkeley, CA, 1989) ed. T. Ratiu, Publ. MSRI, vol. 22, Springer-Verlag, New York, 1991, pp. 9-21. · Zbl 0739.58014
[5] M. R. Adams, J. Harnad, and E. Previato, Isospectral Hamiltonian flows in finite and infinite dimensions I. Generalized Moser systems and moment maps into loop algebras , Comm. Math. Phys. 117 (1988), no. 3, 451-500. · Zbl 0659.58022 · doi:10.1007/BF01223376
[6] M. Adler and P. van Moerbeke, Kowalewski’s asymptotic method, Kac-Moody Lie algebras and regularization , Comm. Math. Phys. 83 (1982), no. 1, 83-106. · Zbl 0491.58017 · doi:10.1007/BF01947073
[7] M. Adler and P. van Moerbeke, The complex geometry of the Kowalewski-Painlevé analysis , Invent. Math. 97 (1989), no. 1, 3-51. · Zbl 0678.58020 · doi:10.1007/BF01850654
[8] 1 M. Adler and P. van Moerbeke, Completely integrable systems, Euclidean Lie algebras, and curves , Adv. in Math. 38 (1980), no. 3, 267-317. · Zbl 0455.58017 · doi:10.1016/0001-8708(80)90007-9
[9] 2 M. Adler and P. van Moerbeke, Linearization of Hamiltonian systems, Jacobi varieties and representation theory , Adv. in Math. 38 (1980), no. 3, 318-379. · Zbl 0455.58010 · doi:10.1016/0001-8708(80)90008-0
[10] V. I. Arnold, Mathematical Methods of Classical Mechanics , Springer Verlag, Berlin, 1978. · Zbl 0386.70001
[11] W. Barth, C. Peters, and A. Van de Ven, Compact Complex Surfaces , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer Verlag, Berlin, 1984. · Zbl 0718.14023
[12] A. Beauville, Jacobiennes des courbes spectrales et systèmes hamiltoniens complètement intégrables , Acta Math. 164 (1990), no. 3-4, 211-235. · Zbl 0712.58031 · doi:10.1007/BF02392754
[13] A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle , J. Differential Geom. 18 (1983), no. 4, 755-782 (1984). · Zbl 0537.53056
[14] R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior Differential Systems , MSRI Publications, vol. 18, Springer-Verlag, New York, 1991. · Zbl 0726.58002
[15] W. L. Chow, On compact complex analytic varieties , Amer. J. Math. 71 (1949), 893-914. JSTOR: · Zbl 0041.48302 · doi:10.2307/2372375
[16] R. Donagi and E. Markman, Cubics, integrable systems and Calabi-Yau three-folds , Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Mathematical Conference Proceedings, vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 199-221. · Zbl 0878.14031
[17] H. M. Farkas and I. Kra, Riemann Surfaces , Graduate Texts in Math., vol. 71, Springer, New York, 1980. · Zbl 0475.30001
[18] J. Fogarty, Algebraic families on an algebraic surface , Amer. J. Math 90 (1968), 511-521. JSTOR: · Zbl 0176.18401 · doi:10.2307/2373541
[19] I. B. Frenkel, A. G. Reiman, and M. A. Semenov-Tian-Sansky, Graded Lie algebras and completely integrable dynamical systems , Dokl. Akad. Nauk SSSR 247 (1979), no. 4, 802-805. · Zbl 0437.58008
[20] P. Griffiths and J. Harris, Principles of Algebraic Geometry , Wiley, New York, 1978. · Zbl 0408.14001
[21] J. Harnad and J. Hurtubise, Generalized tops and moment maps to loop algebras , J. Math. Phys. 32 (1991), no. 7, 1780-1787. · Zbl 0733.70012 · doi:10.1063/1.529241
[22] N. J. Hitchin, The self-duality equations on a Riemann surface , Proc. London Math. Soc. (3) 55 (1987), no. 1, 59-126. · Zbl 0634.53045 · doi:10.1112/plms/s3-55.1.59
[23] N. J. Hitchin, Stable bundles and integrable systems , Duke Math. J. 54 (1987), no. 1, 91-114. · Zbl 0627.14024 · doi:10.1215/S0012-7094-87-05408-1
[24] J. C. Hurtubise, Finite-dimensional coadjoint orbits in loop algebras , Lett. Math. Phys. 30 (1994), no. 2, 99-104. · Zbl 0797.17015 · doi:10.1007/BF00939697
[25] J. C. Hurtubise and N. Kamran, Projective connections, double fibrations, and formal neighbourhoods of lines , Math. Ann. 292 (1992), no. 3, 383-409. · Zbl 0738.58051 · doi:10.1007/BF01444628
[26] A. Iarrobino, Hilbert scheme of points: overview of last ten years , Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. (2), vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 297-320. · Zbl 0646.14002
[27] I. M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations , Russian Math. Surveys 32 (1977), 185-213. · Zbl 0386.35002 · doi:10.1070/RM1977v032n06ABEH003862
[28] I. M. Krichever and S. P. Novikov, Holomorphic bundles over algebraic curves and nonlinear equations , Russian Math. Surveys 32 (1980), 53-79. · Zbl 0548.35100 · doi:10.1070/RM1980v035n06ABEH001974
[29] S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or \(K-3\) surface , Invent. Math. 77 (1984), no. 1, 101-116. · Zbl 0565.14002 · doi:10.1007/BF01389137
[30] 1 A. G. Reyman and M. A. Semenov-Tian-Shansky, Reduction of Hamiltonian systems, affine Lie algebras and Lax equations , Invent. Math. 54 (1979), no. 1, 81-100. · Zbl 0403.58004 · doi:10.1007/BF01391179
[31] 2 A. G. Reyman and M. A. Semenov-Tian-Shansky, Reduction of Hamiltonian systems, affine Lie algebras and Lax equations II , Invent. Math. 63 (1981), no. 3, 423-432. · Zbl 0442.58016 · doi:10.1007/BF01389063
[32] E. K. Sklyanin, Separation of variables in the classical integrable \(\mathrm SL(3)\) magnetic chain , Comm. Math. Phys. 150 (1992), no. 1, 181-191. · Zbl 0764.35106 · doi:10.1007/BF02096572
[33] D. R. D. Scott, Classical functional Bethe ansatz for \(\mathrm SL(N)\): separation of variables for the magnetic chain , J. Math. Phys. 35 (1994), no. 11, 5831-5843. · Zbl 0822.58029 · doi:10.1063/1.530712
[34] P. van Moerbeke, Introduction to algebraic integrable systems and their Painlevé analysis , Theta functions-Bowdoin 1987, Part 1 (Brunswick, ME, 1987), Proc. Sympos Pure Math., vol. 49, Amer. Math. Soc., Providence, RI, 1989, pp. 107-131. · Zbl 0688.70012
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