Faithful Semitoric Systems

Sonja Hohloch ; Silvia Sabatini ; Daniele Sepe ; Margaret Symington

Symmetry, integrability and geometry: methods and applications, Tome 14 (2018)

Résumé

This paper consists of two parts. The first provides a review of the basic properties of integrable and almost-toric systems, with a particular emphasis on the integral affine structure associated to an integrable system. The second part introduces faithful semitoric systems, a generalization of semitoric systems (introduced by Vũ Ngọc and classified by Pelayo and Vũ Ngọc) that provides the language to develop surgeries on almost-toric systems in dimension 4. We prove that faithful semitoric systems are natural building blocks of almost-toric systems. Moreover, we show that they enjoy many of the properties that their (proper) semitoric counterparts do.

Détail

Keywords: completely integrable Hamiltonian systems; almost toric systems; semitoric systems; integral affine geometry; focus-focus singularities.

Bibliographie

[1] Atiyah M. F., “Convexity and commuting Hamiltonians”, Bull. London Math. Soc., 14 (1982), 1–15 | DOI | MR | Zbl

[2] Bolsinov A. V., Fomenko A. T., Integrable Hamiltonian systems. Geometry, topology, classification, Chapman Hall/CRC, Boca Raton, FL, 2004 | DOI | MR | Zbl

[3] Boucetta M., Molino P., “Géométrie globale des systèmes hamiltoniens complètement intégrables: fibrations lagrangiennes singulières et coordonnées action-angle à singularités”, C. R. Acad. Sci. Paris Sér. I Math., 308 (1989), 421–424 | MR | Zbl

[4] Chaperon M., “Normalisation of the smooth focus-focus: a simple proof”, Acta Math. Vietnam, 38 (2013), 3–9 | DOI | MR | Zbl

[5] Crainic M., Fernandes R. L., Martínez-Torres D., Regular Poisson manifolds of compact types (PMCT 2), arXiv: 1603.00064

[6] Cushman R. H., Bates L. M., Global aspects of classical integrable systems, 2nd ed., Birkhäuser/Springer, Basel, 2015 | DOI | MR | Zbl

[7] Dazord P., Delzant T., “Le problème général des variables actions-angles”, J. Differential Geom., 26 (1987), 223–251 | DOI | MR | Zbl

[8] del Hoyo M. L., “Lie groupoids and their orbispaces”, Port. Math., 70 (2013), 161–209, arXiv: 1212.6714 | DOI | MR | Zbl

[9] Delzant T., “Hamiltoniens périodiques et images convexes de l'application moment”, Bull. Soc. Math. France, 116 (1988), 315–339 | DOI | MR | Zbl

[10] Dufour J.-P., Molino P., “Compactification d'actions de $R^n$ et variables action-angle avec singularités”, Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., 20, Springer, New York, 1991, 151–167 | DOI | MR

[11] Duistermaat J. J., “On global action-angle coordinates”, Comm. Pure Appl. Math., 33 (1980), 687–706 | DOI | MR | Zbl

[12] Eliashberg Y., Polterovich L., Symplectic quasi-states on the quadric surface and Lagrangian submanifolds, arXiv: 1006.2501

[13] Eliasson L. H., Hamiltonian systems with Poisson commuting integrals, Ph.D. Thesis, University of Stockholm, 1984

[14] Goldman W., Hirsch M. W., “The radiance obstruction and parallel forms on affine manifolds”, Trans. Amer. Math. Soc., 286 (1984), 629–649 | DOI | MR | Zbl

[15] Guillemin V., Sternberg S., “Convexity properties of the moment mapping”, Invent. Math., 67 (1982), 491–513 | DOI | MR | Zbl

[16] Guillemin V., Sternberg S., “A normal form for the moment map”, Differential Geometric Methods in Mathematical Physics (Jerusalem, 1982), Math. Phys. Stud., 6, Reidel, Dordrecht, 1984, 161–175 | MR

[17] Guillemin V., Sternberg S., Symplectic techniques in physics, 2nd ed., Cambridge University Press, Cambridge, 1990 | MR | Zbl

[18] Harada M., Kaveh K., “Integrable systems, toric degenerations and Okounkov bodies”, Invent. Math., 202 (2015), 927–985, arXiv: 1205.5249 | DOI | MR | Zbl

[19] Hirsch M. W., Differential topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York–Heidelberg, 1976 | DOI | MR | Zbl

[20] Hohloch S., Sabatini S., Sepe D., “From compact semi-toric systems to Hamiltonian $S^1$-spaces”, Discrete Contin. Dyn. Syst., 35 (2015), 247–281, arXiv: 1305.7040 | DOI | MR | Zbl

[21] Hohloch S., Sabatini S., Sepe D., Symington M., From Hamiltonian $S^1$-spaces to compact semitoric systems, in preparation

[22] Hohloch S., Sabatini S., Sepe D., Symington M., Surgeries on faithful semitoric systems, in preparation

[23] Izosimov A. M., “Classification of almost toric singularities of Lagrangian foliations”, Sb. Math., 202 (2011), 1021–1042 | DOI | MR | Zbl

[24] Joyce D., “On manifolds with corners”, Advances in Geometric Analysis, Adv. Lect. Math. (ALM), 21, Int. Press, Somerville, MA, 2012, 225–258, arXiv: 0910.3518 | MR | Zbl

[25] Karshon Y., Lerman E., “Non-compact symplectic toric manifolds”, SIGMA, 11 (2015), 055, 37 pp., arXiv: 0907.2891 | DOI | MR | Zbl

[26] Karshon Y., Tolman S., “Centered complexity one Hamiltonian torus actions”, Trans. Amer. Math. Soc., 353 (2001), 4831–4861, arXiv: math.SG/9911189 | DOI | MR | Zbl

[27] Le Floch Y., Pelayo Á., Vũ Ngọc S., “Inverse spectral theory for semiclassical Jaynes–Cummings systems”, Math. Ann., 364 (2016), 1393–1413, arXiv: 1407.5159 | DOI | MR | Zbl

[28] Leung N. C., Symington M., “Almost toric symplectic four-manifolds”, J. Symplectic Geom., 8 (2010), 143–187, arXiv: math.SG/0312165 | DOI | MR | Zbl

[29] Marle C.-M., “Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique”, Rend. Sem. Mat. Univ. Politec. Torino, 43 (1985), 227–251 | MR | Zbl

[30] Meigniez G., “Submersions, fibrations and bundles”, Trans. Amer. Math. Soc., 354 (2002), 3771–3787 | DOI | MR | Zbl

[31] Miranda E., Zung N. T., “Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems”, Ann. Sci. École Norm. Sup., 37 (2004), 819–839, arXiv: math.SG/0302287 | DOI | MR | Zbl

[32] Moerdijk I., Mrčun J., Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, Cambridge, 2003 | DOI | MR | Zbl

[33] Mrčun J., “An extension of the Reeb stability theorem”, Topology Appl., 70 (1996), 25–55 | DOI | MR | Zbl

[34] Palais R. S., “When proper maps are closed”, Proc. Amer. Math. Soc., 24 (1970), 835–836 | DOI | MR | Zbl

[35] Pelayo Á., Ratiu T. S., Vũ Ngọc S., “Fiber connectivity and bifurcation diagrams of almost toric integrable systems”, J. Symplectic Geom., 13 (2015), 343–386 | DOI | MR | Zbl

[36] Pelayo Á., Ratiu T. S., Vũ Ngọc S., “The affine invariant of proper semitoric integrable systems”, Nonlinearity, 30 (2017), 3993–4028, arXiv: 1307.7516 | DOI | MR | Zbl

[37] Pelayo Á., Tang X., Vũ Ngọc's conjecture on focus-focus fibers with multiple pinched points, arXiv: 1803.00998

[38] Pelayo Á., Vũ Ngọc S., “Semitoric integrable systems on symplectic 4-manifolds”, Invent. Math., 177 (2009), 571–597, arXiv: 0806.1946 | DOI | MR | Zbl

[39] Pelayo Á., Vũ Ngọc S., “Constructing integrable systems of semitoric type”, Acta Math., 206 (2011), 93–125, arXiv: 0903.3376 | DOI | MR | Zbl

[40] Pelayo Á., Vũ Ngọc S., “Hamiltonian dynamics and spectral theory for spin-oscillators”, Comm. Math. Phys., 309 (2012), 123–154, arXiv: 1005.0439 | DOI | MR | Zbl

[41] Sepe D., Vũ Ngọc S., “Integrable systems, symmetries, and quantization”, Lett. Math. Phys., 108 (2018), 499–571, arXiv: 1704.06686 | DOI | MR | Zbl

[42] Sullivan D., Thurston W., “Manifolds with canonical coordinate charts: some examples”, Enseign. Math., 29 (1983), 15–25 | MR | Zbl

[43] Symington M., “Four dimensions from two in symplectic topology”, Topology and Geometry of Manifolds (Athens, GA, 2001), Proc. Sympos. Pure Math., 71, Amer. Math. Soc., Providence, RI, 2003, 153–208, arXiv: math.SG/0210033 | DOI | MR | Zbl

[44] Vũ Ngọc S., “Bohr–Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type”, Comm. Pure Appl. Math., 53 (2000), 143–217, arXiv: math.AP/9810137 | 3.0.CO;2-D class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[45] Vũ Ngọc S., “On semi-global invariants for focus-focus singularities”, Topology, 42 (2003), 365–380, arXiv: math.SG/0208245 | DOI | MR | Zbl

[46] Vũ Ngọc S., Systèmes intégrables semi-classiques: du local au global, Panoramas et Synthèses, 22, Société Mathématique de France, Paris, 2006 | MR

[47] Vũ Ngọc S., “Moment polytopes for symplectic manifolds with monodromy”, Adv. Math., 208 (2007), 909–934, arXiv: math.SG/0504165 | DOI | MR | Zbl

[48] Vũ Ngọc S., Wacheux C., “Smooth normal forms for integrable Hamiltonian systems near a focus-focus singularity”, Acta Math. Vietnam., 38 (2013), 107–122, arXiv: 1103.3282 | DOI | MR | Zbl

[49] Vianna R., “On exotic Lagrangian tori in ${\mathbb{CP}}^2$”, Geom. Topol., 18 (2014), 2419–2476, arXiv: 1305.7512 | DOI | MR | Zbl

[50] Vianna R., “Infinitely many exotic monotone Lagrangian tori in ${\mathbb{CP}}^2$”, J. Topol., 9 (2016), 535–551, arXiv: 1409.2850 | DOI | MR | Zbl

[51] Vianna R., “Infinitely many monotone Lagrangian tori in del Pezzo surfaces”, Selecta Math. (N.S.), 23 (2017), 1955–1996, arXiv: 1602.03356 | DOI | MR | Zbl

[52] Williamson J., “On the algebraic problem concerning the normal forms of linear dynamical systems”, Amer. J. Math., 58 (1936), 141–163 | DOI | MR

[53] Zung N. T., “Symplectic topology of integrable Hamiltonian systems. I Arnold–Liouville with singularities”, Compositio Math., 101 (1996), 179–215, arXiv: math.DS/0106013 | MR | Zbl

[54] Zung N. T., “A note on focus-focus singularities”, Differential Geom. Appl., 7 (1997), 123–130, arXiv: math.DS/0110147 | DOI | MR | Zbl

[55] Zung N. T., “Symplectic topology of integrable Hamiltonian systems. II Topological classification”, Compositio Math., 138 (2003), 125–156, arXiv: math.DG/0010181 | DOI | MR | Zbl