Elliptic K3 surfaces as dynamical models and their Hamiltonian monodromy

Daisuke Tarama

Open Mathematics (2012)

  • Volume: 10, Issue: 5, page 1619-1626
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Daisuke Tarama. "Elliptic K3 surfaces as dynamical models and their Hamiltonian monodromy." Open Mathematics 10.5 (2012): 1619-1626. <http://eudml.org/doc/269278>.

@article{DaisukeTarama2012,
abstract = {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.},
author = {Daisuke Tarama},
journal = {Open Mathematics},
keywords = {K3 dynamical model; Completely integrable system; Lagrangian fibration; Elliptic K3 surface; Monodromy; completely integrable system; elliptic K3 surface; monodromy},
language = {eng},
number = {5},
pages = {1619-1626},
title = {Elliptic K3 surfaces as dynamical models and their Hamiltonian monodromy},
url = {http://eudml.org/doc/269278},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Daisuke Tarama
TI - Elliptic K3 surfaces as dynamical models and their Hamiltonian monodromy
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1619
EP - 1626
AB - 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.
LA - eng
KW - K3 dynamical model; Completely integrable system; Lagrangian fibration; Elliptic K3 surface; Monodromy; completely integrable system; elliptic K3 surface; monodromy
UR - http://eudml.org/doc/269278
ER -

References

top
  1. [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 Zbl1036.14016
  2. [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 Zbl1134.37353
  3. [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 Zbl1109.58027
  4. [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 Zbl0882.58023
  5. [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 Zbl0439.58014
  6. [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. [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. [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 Zbl0402.14014
  9. [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 Zbl0098.13004
  10. [10] Leung N.C., Symington M., Almost toric symplectic four-manifolds, J. Symplectic Geom., 2010, 8(2), 143–187 Zbl1197.53103
  11. [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 Zbl0637.58004
  12. [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. [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 Zbl1202.81100

NotesEmbed ?

top

You must be logged in to post comments.