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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the construction of certain 6-dimensional symplectic manifolds with Hamiltonian circle actions


Author: Hui Li
Journal: Trans. Amer. Math. Soc. 357 (2005), 983-998
MSC (2000): Primary 53D05, 53D20; Secondary 57R17
Published electronically: October 19, 2004
MathSciNet review: 2110428
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Abstract: Let $(M, \omega)$ be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian $S^1$ action such that the fixed point set consists of isolated points or surfaces. Assume dim $H^2(M)<3$. In an earlier paper, we defined a certain invariant of such spaces which consists of fixed point data and twist type, and we divided the possible values of these invariants into six ``types''. In this paper, we construct such manifolds with these ``types''. As a consequence, we have a precise list of the values of these invariants.


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  • [A] M. Atiyah, Convexity and commuting Hamiltonians, Bull. Lond. Math. Soc. 14 (1982), 1-15. MR 0642416 (83e:53037)
  • [DH] J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space and Addendum, Invent. Math. 69, 259-269 (1982); 72, 153-158 (1983). MR 0674406 (84h:58051a); MR 0696693 (84h:58051b)
  • [GS] V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Math. 97, 485-522 (1989). MR 1005004 (90f:58060)
  • [HY] Peichu Hu and Mingze Yang, A theorem of differential mappings of Riemann surfaces, Internat. J. Math. Math. Sci. 17, No. 1 (1994), 189-192. MR 1255241 (94k:30103)
  • [L] H. Li, Semi-free Hamiltonian circle actions on 6-dimensional symplectic manifolds, Trans. Amer. Math. Soc. 355, No. 11, 4543-4568, 2003. MR 1990761 (2004e:53127)
  • [LMc] F. Lalonde and D. McDuff, The classification of ruled symplectic 4-manifolds, Mathematical Research Letters 3, 769-778 (1996). MR 1426534 (98b:57040)
  • [Mc1] D. McDuff, The moment map for circle actions on symplectic manifolds, Journal of Geometry and Physics 5, 149 (1988). MR 1029424 (91c:58042)
  • [Mc2] D. McDuff, The structure of rational and ruled symplectic 4-manifolds, J. Amer. Math. Soc. 3 (1990), 679-712. MR 1049697 (91k:58042)
  • [T] C. Taubes, Seiberg-Witten and Gromov invariants, Geometry and physics (Aarhus, 1995), 591-601, Lecture Notes in Pure and Appl. Math., 184, Dekker, New York, 1997. MR 1423194 (97j:57051)

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Additional Information

Hui Li
Affiliation: Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001, Lisbon, Portugal
Email: hli@math.uiuc.edu, hli@math.ist.utl.pt

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03762-6
PII: S 0002-9947(04)03762-6
Keywords: Symplectic manifold, Hamiltonian $S^1$ action, moment map, symplectic quotient, symplectic submanifolds.
Received by editor(s): May 5, 2003
Published electronically: October 19, 2004
Additional Notes: The author acknowledges the support of the center of Analysis, Geometry, and Dynamical systems in Lisbon, Portugal, where this paper was revised.
Article copyright: © Copyright 2004 American Mathematical Society