This note deals with Lagrangian fibrations of elliptic K3 surfaces and the associated Hamiltonian monodromy. The fibration is constructed through the Weierstraß normal form of elliptic surfaces. There is given an example of K3 dynamical models with the identity monodromy matrix around 12 elementary singular loci.
@article{bwmeta1.element.doi-10_2478_s11533-012-0050-6,
author = {Daisuke Tarama},
title = {Elliptic K3 surfaces as dynamical models and their Hamiltonian monodromy},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1619-1626},
zbl = {06137103},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0050-6}
}
Daisuke Tarama. Elliptic K3 surfaces as dynamical models and their Hamiltonian monodromy. Open Mathematics, Tome 10 (2012) pp. 1619-1626. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0050-6/
[1] Barth W.P., Hulek K., Peters C.A.M., Van de Ven A., Compact Complex Surfaces, 2nd ed., Springer, Berlin-Heidelberg-New York, 2004 | Zbl 1036.14016
[2] Bates L., Cushman R., Complete integrability beyond Liouville-Arnol’d, Rep. Math. Phys., 2005, 56(1), 77–91 http://dx.doi.org/10.1016/S0034-4877(05)80042-4 | Zbl 1134.37353
[3] Bolsinov A.V., Dullin H.R., Veselov A.P., Spectra of sol-manifolds: arithmetic and quantum monodromy, Commun. Math. Phys., 2006, 264, 583–611 http://dx.doi.org/10.1007/s00220-006-1543-6 | Zbl 1109.58027
[4] Cushman R.H., Bates L.M., Global Aspects of Classical Integrable Systems, Birkhäuser, Basel-Boston-Berlin, 1997 http://dx.doi.org/10.1007/978-3-0348-8891-2 | Zbl 0882.58023
[5] Duistermaat J.J., On global action-angle coordinates, Comm. Pure Appl. Math., 1980, 33(6), 687–706 http://dx.doi.org/10.1002/cpa.3160330602 | Zbl 0439.58014
[6] Flaschka H., A remark on integrable Hamiltonian systems, Phy. Lett. A, 1988, 131(9), 505–508 http://dx.doi.org/10.1016/0375-9601(88)90678-0
[7] Hurwitz A., Vorlesungen über Allgemeine Funktionentheorie und Elliptische Funktionen, 5th ed., Berlin-Heidelberg-New York, Springer, 2000 http://dx.doi.org/10.1007/978-3-642-56952-4
[8] Kas A., Weierstrass normal forms and invariants of elliptic surfaces, Trans. Amer. Math. Soc., 1977, 225, 259–266 http://dx.doi.org/10.1090/S0002-9947-1977-0422285-X | Zbl 0402.14014
[9] Kodaira K., On compact analytic surfaces: I, II, III, Ann. of Math., 1960, 71, 111–152; 1963, 77, 563-626; 1963, 78, 1–40 http://dx.doi.org/10.2307/1969881 | Zbl 0098.13004
[10] Leung N.C., Symington M., Almost toric symplectic four-manifolds, J. Symplectic Geom., 2010, 8(2), 143–187 | Zbl 1197.53103
[11] Markushevich D.G., Integrable symplectic structures on compact complex manifolds, Math. USSR-Sb., 1988, 59(2), 459–469 http://dx.doi.org/10.1070/SM1988v059n02ABEH003146 | Zbl 0637.58004
[12] Pjateckiĭ-Šapiro I.I., Šafarevič I.R., A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izvestija, 1971, 5(3), 547–588 http://dx.doi.org/10.1070/IM1971v005n03ABEH001075
[13] Zhilinskií B., Quantum monodromy and pattern formation, 2010, J. Phys. A, 43, #434033 http://dx.doi.org/10.1088/1751-8113/43/43/434033 | Zbl 1202.81100