Number of fixed points for unitary Tⁿ⁻¹-manifold

Abstract

Let M be a 2n-dimensional closed unitary manifold with a Tⁿ⁻¹-action with only isolated fixed points. In this paper, we first prove that the equivariant cobordism class of a unitary Tⁿ⁻¹-manifold M is just determined by the equivariant Chern numbers c_{\omega }^{T^{n-\mathrm{1}}}[M],where ω= (i₁, i₂, ..., i₆) are the multi-indexes for all i₁, i₂, ..., i₆∈\mathbb{N}. Then we show that if Mdoes not bound equivariantly, then the number of fixed points is greater than or equal to \lceil n/\mathrm{6}\rceil +\mathrm{1}, where \lceil n/\mathrm{6}\rceil denotes the minimum integer no less than n/6.

Keywords

Unitary torus manifold / equivariant Chern number / cobordism / localization theorem

References

[1] Borel A, Hirzebruch F. Characteristic classes and homogeneous spaces, I. Amer J Math, 1958, 80(2): 458–538

[2] Borel A, Hirzebruch F. Characteristic classes and homogeneous spaces, II. Amer J Math, 1959, 81(2): 315–382

[3] Buchstaber V M, Krichever I M. Geometry, Topology, and Mathematical Physics. S. P. Novikov’s Seminar: 2006-2007. Amer Math Soc Transl Ser 2, Vol 224. Providence: Amer Math Soc, 2008

[4] Buchstaber V M, Panov T E. Toric Topology. Math Surveys Monogr, Vol 204. Providence: Amer Math Soc, 2015

[5] Cho H W, Kim J H, Park H C. On the conjecture of Kosniowski. Asian J Math, 2012, 16(2): 271–278

[6] Darby A. Torus manifolds in equivariant complex bordism. Topology Appl, 2015, 189: 31–64

[7] Guillemin V, Ginzburg V, Karshon Y. Moment Maps, Cobordisms, and Hamiltonian Group Actions. Math Surveys Monogr, Vol 98. Providence: Amer Math Soc, 2002

[8] Hattori A. S1-actions on unitary manifolds and quasi-ample line bundles. J Fac Sci Univ Tokyo, Sect IA Math, 1985, 31: 433–486

[9] Jang D. Circle actions on almost complex manifolds with 4 fixed points. Math Z, 2019,

[10] Jang D, Blackburn N. Circle actions on almost complex manifolds with isolated fixed points. J Geom Phys, 2017, 119: 187–192

[11] Kosniowski C. Some formulae and conjectures associated with circle actions. In: Koschorke U, Neumann W D, eds. Topology Symposium Siegen 1979: Proceedings of a Symposium Held at the University of Siegen, June 14–19, 1979. Lecture Notes in Math, Vol 788. Berlin: Springer, 1980, 331–339

[12] Kustarev A A. Almost complex circle actions with few fixed points. Russian Math Surveys, 2013, 68(3): 574–576

[13] Li P, Liu K. Some remarks on circle action on manifolds. Math Res Lett, 2011, 18(3): 437–446

[14] Lü Z, Tan Q. Equivariant Chern numbers and the number of fixed points for unitary torus manifolds. Math Res Lett, 2011, 18(6): 1319–1325

[15] Lü Z, Wang W. Equivariant cohomology Chern numbers determine equivariant unitary bordism for torus groups. Algebr Geom Topol, 2018, 18: 4143–4160

[16] Musin O R. Circle actions with two fixed points. Math Notes, 2016, 100: 636–638

[17] Pelayo A, Tolman S. Fixed points of symplectic periodic flows. Ergodic Theory Dynam Systems, 2011, 31(4): 1237–1247

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