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SIX DIMENSIONAL ALMOST COMPLEX TORUS MANIFOLDS WITH EULER NUMBER SIX

  • Donghoon Jang (Department of Mathematics Pusan National University) ;
  • Jiyun Park (Department of Mathematics Pusan National University)
  • Received : 2023.04.11
  • Accepted : 2023.08.29
  • Published : 2024.03.31

Abstract

An almost complex torus manifold is a 2n-dimensional compact connected almost complex manifold equipped with an effective action of a real n-dimensional torus Tn ≃ (S1)n that has fixed points. For an almost complex torus manifold, there is a labeled directed graph which contains information on weights at the fixed points and isotropy spheres. Let M be a 6-dimensional almost complex torus manifold with Euler number 6. We show that two types of graphs occur for M, and for each type of graph we construct such a manifold M, proving the existence. Using the graphs, we determine the Chern numbers and the Hirzebruch χy-genus of M.

Keywords

Acknowledgement

Donghoon Jang was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (2021R1C1C1004158).

References

  1. M. F. Atiyah and R. H. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1-28. https://doi.org/10.1016/0040-9383(84)90021-1 
  2. A. Darby, Torus manifolds in equivariant complex bordism, Topology Appl. 189 (2015), 31-64. https://doi.org/10.1016/j.topol.2015.03.014 
  3. L. Godinho and S. Sabatini, New tools for classifying Hamiltonian circle actions with isolated fixed points, Found. Comput. Math. 14 (2014), no. 4, 791-860. https://doi.org/10.1007/s10208-014-9204-1 
  4. A. Hattori and H. Taniguchi, Smooth S1-action and bordism, J. Math. Soc. Japan 24 (1972), 701-731. https://doi.org/10.2969/jmsj/02440701 
  5. D. Jang, Symplectic periodic flows with exactly three equilibrium points, Ergodic Theory Dynam. Systems 34 (2014), no. 6, 1930-1963. https://doi.org/10.1017/etds.2014.56 
  6. D. Jang, Circle actions on almost complex manifolds with isolated fixed points, J. Geom. Phys. 119 (2017), 187-192. https://doi.org/10.1016/j.geomphys.2017.05.004 
  7. D. Jang, Circle actions on almost complex manifolds with 4 fixed points, Math. Z. 294 (2020), no. 1-2, 287-319. https://doi.org/10.1007/s00209-019-02267-z 
  8. D. Jang, Almost complex torus manifolds-graphs, Hirzebruch genera, and problem of Petrie type, arXiv:2201.00352, 2022. 
  9. D. Jang, Circle actions on four dimensional almost complex manifolds with discrete fixed point sets, Int. Math. Res. Notices (2023), rnad285. https://doi.org/10.1093/imrn/rnad285 
  10. D. Jang, Almost complex torus manifolds - graphs and Hirzebruch genera, Int. Math. Res. Notices 2023 (2023), no. 17, 14594-14609. https://doi.org/10.1093/imrn/rnac237 
  11. D. Jang and S. Tolman, Hamiltonian circle actions on eight-dimensional manifolds with minimal fixed sets, Transform. Groups 22 (2017), no. 2, 353-359. https://doi.org/10.1007/s00031-016-9370-0 
  12. S. Kobayashi, Fixed points of isometries, Nagoya Math. J. 13 (1958), 63-68. https://projecteuclid.org/euclid.nmj/1118800030  https://doi.org/10.1017/S0027763000023497
  13. C. Kosniowski, Applications of the holomorphic Lefschetz formula, Bull. London Math. Soc. 2 (1970), 43-48. https://doi.org/10.1112/blms/2.1.43 
  14. C. Kosniowski, Holomorphic vector fields with simple isolated zeros, Math. Ann. 208 (1974), 171-173. https://doi.org/10.1007/BF01432385 
  15. C. Kosniowski and M. Yahia, Unitary bordism of circle actions, Proc. Edinburgh Math. Soc. (2) 26 (1983), no. 1, 97-105. https://doi.org/10.1017/S001309150002811X 
  16. H. Li, Hamiltonian circle actions with almost minimal isolated fixed points, J. Geom. Phys. 163 (2021), Paper No. 104141, 12 pp. https://doi.org/10.1016/j.geomphys.2021.104141 
  17. H. Li, Hamiltonian circle actions with minimal isolated fixed points, Math. Z. 304 (2023), no. 2, Paper No. 33, 22 pp. https://doi.org/10.1007/s00209-023-03288-5 
  18. M. Masuda, Unitary toric manifolds, multi-fans and equivariant index, Tohoku Math. J. (2) 51 (1999), no. 2, 237-265. https://doi.org/10.2748/tmj/1178224815