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)

Request Permissions   Purchase Content 
 
 

 

Tame circle actions


Authors: Susan Tolman and Jordan Watts
Journal: Trans. Amer. Math. Soc. 369 (2017), 7443-7467
MSC (2010): Primary 53D20; Secondary 53D05, 53B35
DOI: https://doi.org/10.1090/tran/7113
Published electronically: June 13, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the Kähler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.)

Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting and elucidates the key role played by the following fact: the moment image of $ e^t \cdot x$ increases as $ t \in \mathbb{R}$ increases.


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

  • [1] M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1-28. MR 721448, https://doi.org/10.1016/0040-9383(84)90021-1
  • [2] D. Burns, V. Guillemin, and E. Lerman, Kähler cuts, preprint (2002), arXiv:math/021206.
  • [3] Igor V. Dolgachev and Yi Hu, Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 5-56. MR 1659282
  • [4] 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 (1982), no. 2, 259-268. MR 674406, https://doi.org/10.1007/BF01399506
  • [5] Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, Mathematical Surveys and Monographs, vol. 98, American Mathematical Society, Providence, RI, 2002. MR 1929136
  • [6] Leonor Godinho, Blowing up symplectic orbifolds, Ann. Global Anal. Geom. 20 (2001), no. 2, 117-162. MR 1857175, https://doi.org/10.1023/A:1011628628835
  • [7] William Graham, Logarithmic convexity of push-forward measures, Invent. Math. 123 (1996), no. 2, 315-322. MR 1374203, https://doi.org/10.1007/s002220050029
  • [8] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515-538. MR 664118, https://doi.org/10.1007/BF01398934
  • [9] V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Math. 97 (1989), no. 3, 485-522. MR 1005004, https://doi.org/10.1007/BF01388888
  • [10] P. Heinzner and F. Loose, Reduction of complex Hamiltonian $ G$-spaces, Geom. Funct. Anal. 4 (1994), no. 3, 288-297. MR 1274117, https://doi.org/10.1007/BF01896243
  • [11] N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), no. 4, 535-589. MR 877637
  • [12] Frances Clare Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984. MR 766741
  • [13] Eugene Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247-258. MR 1338784, https://doi.org/10.4310/MRL.1995.v2.n3.a2
  • [14] Eugene Lerman and Susan Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997), no. 10, 4201-4230. MR 1401525, https://doi.org/10.1090/S0002-9947-97-01821-7
  • [15] Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. MR 1698616
  • [16] Reyer Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2) 141 (1995), no. 1, 87-129. MR 1314032, https://doi.org/10.2307/2118628
  • [17] S. Tolman, Non-Hamiltonian actions with isolated fixed points, Invent. Math., to appear.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53D20, 53D05, 53B35

Retrieve articles in all journals with MSC (2010): 53D20, 53D05, 53B35


Additional Information

Susan Tolman
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: stolman@math.uiuc.edu

Jordan Watts
Affiliation: Department of Mathematics, University of Colorado Boulder, Boulder, Colorado 80309
Email: jordan.watts@colorado.edu

DOI: https://doi.org/10.1090/tran/7113
Keywords: Tamed symplectic form, symplectic reduction, blow-up, symplectic cutting, holomorphic action, Hamiltonian action, K\"ahler manifold, moment map
Received by editor(s): September 29, 2015
Received by editor(s) in revised form: June 17, 2016
Published electronically: June 13, 2017
Additional Notes: The first author was partially supported by National Science Foundation Grant DMS-1206365
The second author thanks the University of Illinois at Urbana-Champaign for their support. Moreover, this manuscript was significantly improved by suggestions from an anonymous referee; the authors extend their thanks.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society