The rigidity of Dolbeault-type operators and symplectic circle actions

Li, Ping

doi:10.1090/S0002-9939-2011-11067-0

The rigidity of Dolbeault-type operators and symplectic circle actions
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Proc. Amer. Math. Soc. 140 (2012), 1987-1995 Request permission

Abstract:

Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce many identities concerning the weights on the fixed points. In particular, it gives a criterion to determine whether or not a symplectic circle action with isolated fixed points is Hamiltonian. As applications, we simplify the proofs of some known results related to symplectic circle actions, due to Godinho, Tolman-Weitsman and Pelayo-Tolman, and generalize some of them to more general cases.

References

Michael Atiyah and Friedrich Hirzebruch, Spin-manifolds and group actions, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 18–28. MR 0278334
M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
Raoul Bott and Clifford Taubes, On the rigidity theorems of Witten, J. Amer. Math. Soc. 2 (1989), no. 1, 137–186. MR 954493, DOI 10.1090/S0894-0347-1989-0954493-5
K. È. Fel′dman, Hirzebruch genera of manifolds supporting a Hamiltonian circle action, Uspekhi Mat. Nauk 56 (2001), no. 5(341), 187–188 (Russian); English transl., Russian Math. Surveys 56 (2001), no. 5, 978–979. MR 1892568, DOI 10.1070/RM2001v056n05ABEH000446
Theodore Frankel, Fixed points and torsion on Kähler manifolds, Ann. of Math. (2) 70 (1959), 1–8. MR 131883, DOI 10.2307/1969889
Leonor Godinho, On certain symplectic circle actions, J. Symplectic Geom. 3 (2005), no. 3, 357–383. MR 2198781, DOI 10.4310/JSG.2005.v3.n3.a3
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
Akio Hattori and Hajime Taniguchi, Smooth $S^{1}$-action and bordism, J. Math. Soc. Japan 24 (1972), 701–731. MR 309134, DOI 10.2969/jmsj/02440701
F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. MR 0202713
Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and modular forms, Aspects of Mathematics, E20, Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils-Peter Skoruppa and by Paul Baum. MR 1189136, DOI 10.1007/978-3-663-14045-0
C. Kosniowski, Applications of the holomorphic Lefschetz formula, Bull. London Math. Soc. 2 (1970), 43–48. MR 261636, DOI 10.1112/blms/2.1.43
Czes Kosniowski, Holomorphic vector fields with simple isolated zeros, Math. Ann. 208 (1974), 171–173. MR 341501, DOI 10.1007/BF01432385
Czes Kosniowski, Some formulae and conjectures associated with circle actions, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 331–339. MR 585666
Czes Kosniowski and Mahgoub Yahia, Unitary bordism of circle actions, Proc. Edinburgh Math. Soc. (2) 26 (1983), no. 1, 97–105. MR 695647, DOI 10.1017/S001309150002811X
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
Gheorghe Lusztig, Remarks on the holomorphic Lefschetz numbers, Analyse globale, (Sém. Math. Supérieures, No. 42, 1969) Presses Univ. Montréal, Montreal, Que., 1971, pp. 193–204. MR 0380821
Dusa McDuff, The moment map for circle actions on symplectic manifolds, J. Geom. Phys. 5 (1988), no. 2, 149–160. MR 1029424, DOI 10.1016/0393-0440(88)90001-0
A. Pelayo, S. Tolman: Fixed points of symplectic periodic flows, Ergodic Theory and Dynamical Systems 31 (2011), 1237-1247.
Susan Tolman and Jonathan Weitsman, On semifree symplectic circle actions with isolated fixed points, Topology 39 (2000), no. 2, 299–309. MR 1722020, DOI 10.1016/S0040-9383(99)00011-7