Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Frankel's theorem in the symplectic category


Author: Min Kyu Kim
Journal: Trans. Amer. Math. Soc. 358 (2006), 4367-4377
MSC (2000): Primary 53D05, 53D20; Secondary 55Q05, 57R19
DOI: https://doi.org/10.1090/S0002-9947-06-03844-X
Published electronically: January 24, 2006
MathSciNet review: 2231381
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that if an $ (n-1)$-dimensional torus acts symplectically on a $ 2n$-dimensional symplectic manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel's theorem which says that a Kähler circle action has a fixed point if and only if it is Hamiltonian. The case of $ n=2$ is the well-known theorem by McDuff.


References [Enhancements On Off] (What's this?)

  • [A] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1-15. MR 0642416 (83e:53037)
  • [D] T. Delzant, Hamiltoniens periodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), 315-339. MR 0984900 (90b:58069)
  • [Fe] K. E. Feldman, Hirzebruch genera of manifolds supporting a Hamiltonian circle action, Russian Math. Surveys 56 (2001), no. 5, 978-979. MR 1892568 (2003a:57062)
  • [Fr] T. Frankel, Fixed points on Kähler manifolds, Ann. Math. 70 (1959), 1-8. MR 0131883 (24:A1730)
  • [Fu] W. Fulton, Introduction to Toric Varieties, Ann. Math. Studies 131, Princeton University Press, 1993. MR 1234037 (94g:14028)
  • [GS] V. Guillemin, S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491-513. MR 0664117 (83m:58037)
  • [KT1] Y. Karshon, S. Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc. 353 (2001), no. 12, 4831-4861. MR 1852084 (2002g:53145)
  • [KT2] Y. Karshon, S. Tolman, Complete invariants for Hamiltonian torus actions with two dimensional quotients, J. Symplectic Geom. 2 (2003), no. 1, 25-82. MR 2128388
  • [LT] E. Lerman, S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997), no. 10, 4201-4230. MR 1401525 (98a:57043)
  • [L] H. Li, $ \pi_1$ of Hamiltonian $ S^1$ manifolds, Proc. Amer. Math. Soc. 131, no. 11, 3579-3582. MR 1991771 (2004b:53145)
  • [M] D. McDuff, The moment map for circle actions on symplectic manifolds, J. Geom. Phys. 5 (1988), no. 2, 149-160. MR 1029424 (91c:58042)
  • [MS] D. McDuff, D. Salamon, Introduction to Symplectic Topology, 2nd ed., Oxford University Press, 1998. MR 1698616 (2000g:53098)
  • [O] K. Ono, Equivariant projective imbedding theorem for symplectic manifolds, J. Fac. Sci. Univ. Tokyo Sect. IA Math. (2) 35 (1988), no. 2, 381-392. MR 0945884 (89k:58103)
  • [R] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, 1991. MR 1157815 (92k:46001)
  • [Sa] I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A., 42 (1956), 359-363. MR 0079769 (18:144a)
  • [Sl] P. Sleewaegen, On generalized moment maps for symplectic compact group actions, math. SG/0304487.
  • [SL] R. Sjamaar, E. Lerman, Stratified symplectic spaces and reduction, Ann. Math. 134 (1991), no. 2, 375-422. MR 1127479 (92g:58036)
  • [T] S. Tolman, Examples of non-Kähler Hamiltonian torus actions, Invent. Math. 131 (1998), no. 2, 299-310. MR 1608575 (2000d:53129)
  • [TW] S. Tolman, J. Weitsman, On semifree symplectic circle actions with isolated fixed points, Topology 39 (2000), no. 2, 299-309. MR 1722020 (2000k:53074)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53D05, 53D20, 55Q05, 57R19

Retrieve articles in all journals with MSC (2000): 53D05, 53D20, 55Q05, 57R19


Additional Information

Min Kyu Kim
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, 373-1, Kusong-Dong, Yusong-Gu, Taejon, 305-701, Korea
Address at time of publication: School of Math, Korea Institute for Advanced Study, Cheongnyangni 2-dong, Dongdaemun-gu, Seoul, 130-722, Republic of Korea
Email: minkyu@kaist.ac.kr, mkkim@kias.re.kr

DOI: https://doi.org/10.1090/S0002-9947-06-03844-X
Keywords: Symplectic geometry, symplectic action, Hamiltonian action
Received by editor(s): April 6, 2004
Received by editor(s) in revised form: August 11, 2004
Published electronically: January 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society