Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Centered complexity one Hamiltonian torus actions

Author(s): Yael Karshon; Susan Tolman
Journal: Trans. Amer. Math. Soc. 353 (2001), 4831-4861.
MSC (2000): Primary 53D20; Secondary 53D35
Posted: July 30, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract:

We consider symplectic manifolds with Hamiltonian torus actions which are ``almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are ``centered" and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmannians $\operatorname{Gr}^+(2,\mathbb R^5)$ and $\operatorname{Gr}^+(2,\mathbb R^6)$ by two equal symplectic balls.

References:

[AH]
K. Ahara and A. Hattori, $4$-dimensional symplectic $S^1$-manifolds admitting moment map, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 38 (1991), 251-298. MR 93b:58048

[At]
M. Atiyah, Convexity and commuting hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15. MR 83e:53037

[Au1]
M. Audin, Hamiltoniens périodiques sur les variétés symplectiques compactes de dimension 4, Géométrie symplectique et mécanique, Proceedings 1988, C. Albert ed., Springer Lecture Notes in Math. 1416 (1990). MR 91f:57013

[Au2]
M. Audin, The topology of torus actions on symplectic manifolds, Progress in Mathematics 93, Birkhäuser Verlag, Basel, 1991. MR 92m:57046

[BB]
A. Bialynicki-Birula, Remarks on the action of an algebraic torus on $k^n$. II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15, (1967), 123-125. MR 35:6666

[Bi1]
P. Biran, A stability property of symplectic packing, Invent. Math. 136 (1999), 123-155. MR 2000b:57039

[Bi2]
P. Biran, Symplectic packing in dimension 4, GAFA, Geom. Fun. Anal. 7 (1997), 420-437. MR 98i:57057

[BT]
R. Bott and L. W. Tu, Differential forms in algebraic topology, Springer-Verlag, 1982. MR 83i:57016

[De1]
T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), 315-339. MR 90b:58069

[De2]
T. Delzant, Classification des actions hamiltoniennes complèment intégrables de rang deux, Ann. Global Anal. Geom. 8 (1990), 87-112. MR 92f:58078

[DH]
J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69, 259-268 (1982); Addendum, Invent. Math. 72 (1983), 153-158. MR 84h:58051a MR 84h:58051b

[FK]
K.-H. Fieseler and L. Kaup, On the geometry of affine algebraic $\mathbb C^*$-surfaces, in: Problems in the theory of surfaces and their classification (Cortona, 1988), 111-140, Sympos. Math., XXXII, Academic Press, London, 1991. MR 95c:14065

[F]
R. Fintushel, Circle actions on simply connected 4-manifolds, Trans. Amer. Math. Soc. 230 (1977), 147-171, and Classification of circle actions on 4-manifolds, Trans. Amer. Math. Soc. 242 (1978), 377-390. MR 56:16659; MR 81e:57036

[GSj]
V. Guillemin and R. Sjamaar, Lecture at Mass. Inst. Tech., 1995.

[GS1]
V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, I, Invent. Math. 67 (1982), 491-513. MR 83m:58037

[GS2]
V. Guillemin and S. Sternberg, A normal form for the moment map, Differential geometric methods in mathematical physics, (S. Sternberg, Ed.) Reidel, Dordrecht, Holland, 1984. MR 86d:58033

[GS2]
V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Math. 97 (1989), 485-522. MR 90f:58060

[HS]
A. Haefliger and E. Salem, Actions of tori on orbifolds, Ann. Global Anal. Geom. 9 (1991), 37-59. MR 92f:57047

[I]
P. Iglesias, Les $\operatorname{SO}(3)$-variétés symplectiques et leurs classification en dimension 4, Bull. Soc. Math. France 119 (1991), 371-396. MR 92i:57023

[K1]
Y. Karshon, Appendix to: Symplectic packings and algebraic geometry by D. McDuff and L. Polterovich, Invent. Math. 115, 431-434 (1994). MR 95a:58042

[K2]
Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc. 672 (1999). MR 2000c:53113

[Kn]
F. Knop, private communication.

[Ko]
A. Kosinski, Differential manifolds, Pure and Applied Mathematics 138, Academic Press, Inc., Boston, MA (1993). MR 95b:57001

[Kl]
J. L. Koszul, Sur certains groupes de transformations de Lie, Colloque international du Centre National de la Recherche Scientifique 52 (1953), 137-141. MR 15:600g

[LMTW]
E. Lerman, E. Meinrenken, S. Tolman, and C. Woodward, Nonabelian convexity by symplectic cuts, Topology 37 (1998), 245-259. MR 99a:58069

[LT]
E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc., 349 (1997), no. 10, 4201-4230. MR 98a:57043

[LV]
D. Luna and Th. Vust, Plongements d'espaces homogènes, Comment. Math. Helv. 58 (1983), 186-245. MR 85a:14035

[M]
C. M. Marle, Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique, Rendiconti del Seminario Matematico 43 (1985), 227-251. MR 88a:58075

[McD]
D. McDuff, The moment map for circle actions on symplectic manifolds, J. Geom. Phys. 5 (1988), 149-160. MR 91c:58042

[McD]
D. McDuff, Remarks on the uniqueness of symplectic blowing up, in: Symplectic Geometry, London Math. Soc. Lecture Note Ser. 192, 157-167, Cambridge Univ. Press, Cambridge, 1993. MR 95h:57028

[MS]
E. Meinrenken and R. Sjamaar, Singular reduction and quantization, Topology 38 (1999), 699-762. MR 2000f:53114

[Mo]
J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294. MR 32:409

[KKMS]
G. Kempf, F. F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal embeddings. I. Lecture Notes in Math. 339, Springer-Verlag, Berlin-New York, 1973. MR 49:299

[OR]
P. Orlik and F. Raymond, Actions of the torus on $4$-manifolds. I and II, Trans. Amer. Math. Soc. 152 (1970), 531-559, and Topology 13 (1974), 89-12. MR 42:3808; MR 50:1274

[OW]
P. Orlik and P. Wagreich, Algebraic surfaces with $k^*$-actions, Acta Math. 138 (1977), 43-81. MR 57:336

[R]
J. Rynes, Nonsingular affine $k^*$-surfaces, Trans. Amer. Math. Soc. 332 (1992), 889-921. MR 92j:14061

[Sch1]
G. W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63-68. MR 51:6870

[Sch2]
G. W. Schwarz, Lifting smooth homotopies of orbit spaces, I.H.E.S. Publ. Math. 51 (1980), 37-135. MR 81h:57024

[Sj]
R. Sjamaar, Convexity properties of the moment mapping re-examined, Adv. Math. 138 (1998), 46-91. MR 2000a:53148

[T1]
D. A. Timashëv, $G$-manifolds of complexity $1$, (Russian) Uspekhi Mat. Nauk 51 (1996), no. 3 (309), 213-214; English translation: Russian Math. Surveys 51 (1996), 567-568. CMP 96:17

[T2]
D. A. Timashëv, Classification of $G$-manifolds of complexity $1$, (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), 127-162; English translation: Izv. Math. 61 (1997), 363-397. MR 98h:14058

[To]
S. Tolman, Examples of non-Kähler Hamiltonian torus actions, Invent. Math., 131 (1998), 299-310. MR 2000d:53129

[Tr]
L. Traynor Symplectic packing constructions, J. Diff. Geom. 42 (1995), 411-429. MR 96k:53046

[W1]
A. Weinstein, Lectures on Symplectic Manifolds, CBMS Reg. Conf. Ser. in Math. 29, 1977. MR 57:4244; Corrected reprint MR 82b:58039

[MW]
Merriam-Webster's Collegiate Dictionary, http://www.m-w.com/cgi-bin/dictionary

[W]
C. Woodward, The classification of transversal multiplicity-free group actions, Ann. Global Anal. Geom. 14 (1996), 3-42. MR 97a:58067

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53D20, 53D35

Retrieve articles in all Journals with MSC (2000): 53D20, 53D35

Additional Information:

Yael Karshon
Affiliation: Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Email: karshon@math.huji.ac.il

Susan Tolman
Affiliation: Department of Mathematics, Univiversity of Illinois, Urbana, Illinois 61801
Email: stolman@math.uiuc.edu

DOI: 10.1090/S0002-9947-01-02799-4
PII: S 0002-9947(01)02799-4
Received by editor(s): May 9, 2000
Posted: July 30, 2001
Additional Notes: Y. Karshon was partially supported by NSF grant DMS-9404404 during earlier work on this project, and by M.S.R.I. during the fall of 1999
S. Tolman is partially supported by a Sloan fellowship and by NSF grant DMS-980305. The collaboration is partially supported by the United States Israel Binational Science Foundation, grant number 96-210
Copyright of article: Copyright 2001, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google