Symplectic torus actions with coisotropic principal orbits

Duistermaat, Johannes Jisse; Pelayo, Alvaro

Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2239-2327.

Dans cet article nous donnons une classification complète des actions symplectiques d’un tore $T$ sur une variété compacte connexe symplectique $(M,\sigma )$ pour laquelle une, et donc toute orbite principale est une variété coïsotrope de $(M,\sigma )$. Cela veut dire que nous construisons un modèle explicite, défini en termes de certains invariants de la variété, l’action torique et de la forme symplectique.

Pour traiter des actions symplectiques qui ne sont pas hamiltoniennes, nous développons des techniques nouvelles, étendant la théorie d’Atiyah, Guillemin-Sternberg, Delzant et Benoist. En particulier, nous démontrons qu’il y a une notion bien définie de champs de vecteurs constants sur l’espace des orbites $M/T$. En utilisant une généralisation du théorème de Tietze-Nakayama à ce que nous appelons aussi espaces $V$-parallèles, nous obtenons que $M/T$ est isomorphe au produit cartésien d’un polytope de Delzant avec un tore.

Nous construisons alors les champs de vecteurs spéciaux dans $M$ qui se projettent sur les champs de vecteurs constants sur $M/T$, à l’aide desquels le modèle de la variété symplectique avec action torique est défini.

In this paper we completely classify symplectic actions of a torus $T$ on a compact connected symplectic manifold $(M,\sigma )$ when some, hence every, principal orbit is a coisotropic submanifold of $(M,\sigma )$. That is, we construct an explicit model, defined in terms of certain invariants, of the manifold, the torus action and the symplectic form. The invariants are invariants of the topology of the manifold, of the torus action, or of the symplectic form.

In order to deal with symplectic actions which are not Hamiltonian, we develop new techniques, extending the theory of Atiyah, Guillemin-Sternberg, Delzant, and Benoist. More specifically, we prove that there is a well-defined notion of constant vector fields on the orbit space $M/T$. Using a generalization of the Tietze-Nakajima theorem to what we call $V$-parallel spaces, we obtain that $M/T$ is isomorphic to the Cartesian product of a Delzant polytope with a torus.

We then construct special lifts of the constant vector fields on $M/T$, in terms of which the model of the symplectic manifold with the torus action is defined.

Classification : 53D35, 35J05, 35J10, 17B30, 22E25
Keywords: Symplectic, torus actions, coisotropic orbits, classification
Mot clés : symplectique, actions toriques, orbites coïsotropes, classification

[1] Atiyah, M. Convexity and commuting Hamiltonians, Bull. London Math. Soc., Volume 14 (1982), pp. 1-15 | DOI | MR | Zbl

[2] Audin, M. Torus Actions on Symplectic Manifolds, Progress in Mathematics, 93, Birkhäuser Verlag, Basel, 2004 (Second revised edition) | MR | Zbl

[3] Auslander, L. The structure of complete locally affine manifolds, Topology, Volume 3 (1964), pp. 131-139 | DOI | MR | Zbl

[4] Auslander, L.; Markus, L. Holonomy of flat affinely connected manifolds, Ann. of Math., Volume 62 (1955), pp. 139-151 | DOI | MR | Zbl

[5] Benoist, Y. Correction to “Actions symplectiques de groupes compacts” (http://www.dma.ens.fr/~benoist) | Zbl

[6] Benoist, Y. Actions symplectiques de groupes compacts, Geometriae Dedicata, Volume 89 (2002), pp. 181-245 | DOI | MR | Zbl

[7] Benson, C.; Gordon, C. S. Kähler and symplectic structures on nilmanifolds, Topology, Volume 27 (1988), pp. 513-518 | DOI | MR | Zbl

[8] Bott, R.; Tu, L. W. Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, 82, Springer-Verlag, New York, Heidelberg, Berlin, 1982 | MR | Zbl

[9] Cartan, É Sur les nombres de Betti des espaces de groupes clos, Œuvres, partie I, vol. 2, 999–1001, Volume 187, C.R. Acad. Sc., 1928, pp. 196-198 | JFM

[10] Danilov, V. I. The geometry of toric varieties, Russ. Math. Surveys, Volume 33 (1978) no. 2, pp. 97-154 from Uspekhi Mat. Nauk SSSR, 33, 2 (1978) 85–134 | DOI | MR | Zbl

[11] Delzant, T. Hamiltoniens périodiques et image convex de l’application moment, Bull. Soc. Math. France, Volume 116 (1988), pp. 315-339 | EuDML | Numdam | Zbl

[12] Duistermaat, J. J. Equivariant cohomology and stationary phase, Contemp. Math., Volume 179 (1994), pp. 45-62 | MR | Zbl

[13] Duistermaat, J. J. The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Birkhäuser, Boston, 1996 | MR | Zbl

[14] Duistermaat, J. J; Heckman, G. J. On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math., Volume 69 (1982), pp. 259-268 Addendum in: Invent. Math., 72, (1983) 153–158 | DOI | EuDML | MR | Zbl

[15] Duistermaat, J. J; Kolk, J. A. C. Lie Groups, Universitext, Springer, Berlin, 2000 | MR | Zbl

[16] Fernández, M.; Gotay, M. J.; Gray, A. Compact parallelizable four dimensional symplectic and complex manifolds, Proc. Amer. Math. Soc., Volume 103 (1988), pp. 1209-1212 | DOI | MR | Zbl

[17] Giacobbe, A. Convexity of multi-valued momentum maps, Geometriae Dedicata, Volume 111 (2005), pp. 1-22 | DOI | MR | Zbl

[18] Ginzburg, V. L. Some remarks on symplectic actions of compact groups, Math. Z., Volume 210 (1992), pp. 625-640 | DOI | EuDML | MR | Zbl

[19] Greenberg, M. Lectures on Algebraic Topology, W.A. Benjamin, New York, Amsterdam, 1967 | MR | Zbl

[20] Guillemin, V. Moment Maps and Combinatorial Invariants of Hamiltonian $T^n$-Spaces, 122, Progress in Mathematics (Boston, Mass.), Boston, Basel, Berlin, 1994 | MR | Zbl

[21] Guillemin, V.; Lerman, E.; Sternberg, S. Symplectic Fibrations and Multiplicity Diagrams, Cambridge Univ. Press., Cambridge, 1996 | MR | Zbl

[22] Guillemin, V.; Sternberg, S. Convexity properties of the moment mapping, Invent. Math., Volume 67 (1982), pp. 491-513 | DOI | EuDML | MR | Zbl

[23] Guillemin, V.; Sternberg, S. Multiplicity-free spaces, J. Diff. Geom., Volume 19 (1984), pp. 31-56 | MR | Zbl

[24] Guillemin, V.; Sternberg, S. Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984 | MR | Zbl

[25] Haefliger, A.; Salem, É Actions of tori on orbifolds, Ann. Global Anal. Geom., Volume 9 (1991), pp. 37-59 | DOI | MR | Zbl

[26] Hungerford, T. W. Algebra, Springer-Verlag, New York, 1974 (New York etc.: Holt, Rinehart and Winston, Inc.) | MR | Zbl

[27] Karshon, Y. Periodic Hamiltonian flows on four-dimensional manifolds, Memoirs Amer. Math. Soc., Volume 141 (1999) no. 672, pp. viii+71 pp. | MR | Zbl

[28] Karshon, Y.; Tolman, S. Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc., Volume 353 (2001) no. 12, pp. 4831-4861 | DOI | MR | Zbl

[29] Klee, V. Convex sets in linear spaces, Duke Math. J., Volume 18 (1951), pp. 443-466 | DOI | MR | Zbl

[30] Kodaira, K. On the structure of compact analytic surfaces, I, Amer. J. Math., Volume 86 (1964), pp. 751-798 | DOI | MR | Zbl

[31] Kogan, M. On completely integrable systems with local torus actions, Ann. Global Anal. Geom., Volume 15 (1997) no. 6, pp. 543-553 | DOI | MR | Zbl

[32] Koszul, J. L. Sur certains groupes de transformations de Lie, Colloques Int. Centre Nat. Rech. Sci., Volume 52 (1953), pp. 137-141 (Géométrie Différentielle) | MR | Zbl

[33] Lerman, E.; Tolman, S. Hamiltonian torus actions on symplectic orbifolds, Trans. Amer. Math. Soc., Volume 349 (1997), pp. 4201-4230 | DOI | MR | Zbl

[34] Leung, N. C.; Symington, M. Almost toric symplectic four-manifolds, 2003 (arXiv:math.SG/0312165v1) | Zbl

[35] MacLane, S. Categories for the Working Mathematician, Graduate Texts in Mathematics, 5, Springer-Verlag, New York, 1971, 1998 | MR | Zbl

[36] Marle, C. M. Classification des actions hamiltoniennes au voisinage d’une orbite, C. R. Acad. Sci. Paris Sér. I Math., Volume 299 (1984), pp. 249-252 Modèle d’action hamiltonienne d’un groupe de Lie sur une variété symplectique. Rend. Sem. Mat. Univ. Politec. Torino, 43 (1985) 227–251 | Zbl

[37] Mather, J. N. Stability of $C^{\infty }$ mappings: II. Infinitesimal stability implies stability, Ann. of Math., Volume 89 (1969), pp. 254-291 | DOI | MR | Zbl

[38] McDuff, D. The moment map for circle actions on symplectic manifolds, J. Geom. Phys., Volume 5 (1988) no. 2, pp. 149-160 | DOI | MR | Zbl

[39] McDuff, D.; Salamon, D. Introduction to Symplectic Topology. 2nd ed., Oxford Mathematical Monographs. New York, NY: Oxford University Press, 1998 | MR | Zbl

[40] Mukherjee, G. Transformation Groups. Symplectic Torus Actions and Toric Manifolds, Hindustan Book Agency, New Delhi, 2005 (With contributions by C. Allday, M. Masuda, and P. Sankeran) | MR | Zbl

[41] Nakajima, S. Über konvexe Kurven und Flächen., Tôhoku Math. J., Volume 29 (1928), pp. 227-230 | JFM

[42] Novikov, S. P. The Hamiltonian formalism and a multivalued analogue of Morse theory, Russ. Math. Surveys, Volume 37 (1982) no. 5, pp. 1-56 | DOI | MR | Zbl

[43] Orlik, P.; Raymond, F. Actions of the torus on 4-manifolds, I, Trans. Amer. Math. Soc., Volume 152 (1970), pp. 531-559 II, Topology, 13 (1974), 89–112 | MR | Zbl

[44] Ortega, J.-P.; Ratiu, T. S. A symplectic slice theorem, Lett. Math. Phys., Volume 59 (2002), pp. 81-93 | DOI | MR | Zbl

[45] Ortega, J.-P.; Ratiu, T. S. Momentum Maps and Hamiltonian Reduction, Progress in Mathematics (Boston, Mass.), 222, Birkhäuser, Boston, MA, 2004 | MR | Zbl

[46] Palais, R. S.; Stewart, T. E. Torus bundles over a torus, Proc. Amer. Math. Soc., Volume 12 (1961), pp. 26-29 | DOI | MR | Zbl

[47] Pao, P. S. The topological structure of 4-manifolds with effective torus actions, I, Trans. Amer. Math. Soc., Volume 227 (1977), pp. 279-317 II, Ill. J.Math., 21 (1977), 883–894 | MR | Zbl

[48] Pelayo, A. Symplectic actions of two-tori on four-manifolds (ArXiv: Math.SG/0609848) | Zbl

[49] Rockafellar, R. T. Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970 | MR | Zbl

[50] Symington, M. Four dimensions from two in symplectic topology, Proc. Sympos. Pure Math., Volume 71, Amer. Math. Soc., Providence, RI, Topology and geometry of manifolds (Athens, GA, 2001) (2003), pp. 153-208 | MR | Zbl

[51] Thurston, W. P. Some examples of symplectic manifolds, Proc. Amer. Math. Soc., Volume 55 (1976), pp. 467-468 | MR | Zbl

[52] Tietze, H. Über Konvixität im Kleinen und im Großen und über gewisse den Punkten einer Menge zugeordete Dimensionszahlen, Math. Z., Volume 28 (1928), pp. 697-707 | DOI | EuDML | JFM | MR

[53] Whitney, H. Differentiable even functions, Duke Math. J., Volume 10 (1943), pp. 159-160 | DOI | MR | Zbl