Tame circle actions

Tolman, Susan; Watts, Jordan

doi:10.1090/tran/7113

Tame circle actions
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by Susan Tolman and Jordan Watts PDF

Trans. Amer. Math. Soc. 369 (2017), 7443-7467 Request permission

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

M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. MR 721448, DOI 10.1016/0040-9383(84)90021-1
D. Burns, V. Guillemin, and E. Lerman, Kähler cuts, preprint (2002), arXiv:math/021206.
Igor V. Dolgachev and Yi Hu, Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 5–56. With an appendix by Nicolas Ressayre. MR 1659282, DOI 10.1007/BF02698859
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, DOI 10.1007/BF01399506
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. Appendix J by Maxim Braverman. MR 1929136, DOI 10.1090/surv/098
Leonor Godinho, Blowing up symplectic orbifolds, Ann. Global Anal. Geom. 20 (2001), no. 2, 117–162. MR 1857175, DOI 10.1023/A:1011628628835
William Graham, Logarithmic convexity of push-forward measures, Invent. Math. 123 (1996), no. 2, 315–322. MR 1374203, DOI 10.1007/s002220050029
V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515–538. MR 664118, DOI 10.1007/BF01398934
V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Math. 97 (1989), no. 3, 485–522. MR 1005004, DOI 10.1007/BF01388888
P. Heinzner and F. Loose, Reduction of complex Hamiltonian $G$-spaces, Geom. Funct. Anal. 4 (1994), no. 3, 288–297. MR 1274117, DOI 10.1007/BF01896243
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, DOI 10.1007/BF01214418
Frances Clare Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984. MR 766741, DOI 10.2307/j.ctv10vm2m8
Eugene Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247–258. MR 1338784, DOI 10.4310/MRL.1995.v2.n3.a2
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, DOI 10.1090/S0002-9947-97-01821-7
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
Reyer Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2) 141 (1995), no. 1, 87–129. MR 1314032, DOI 10.2307/2118628
S. Tolman, Non-Hamiltonian actions with isolated fixed points, Invent. Math., to appear.