Symplectic geometry
Remark on the Betti numbers for Hamiltonian circle actions
Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 113-117.

In this paper, we establish a certain inequality in terms of Betti numbers of a closed Hamiltonian S 1 -manifold with isolated fixed points.

Dans cet article, nous établissons une certaine inégalité en termes de nombres de Betti d’une S 1 -variété hamiltonienne avec des points fixes isolés.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.127
Classification: 53D20, 53D05
Cho, Yunhyung 1

1 Department of Mathematics Education, Sungkyunkwan University, Seoul, Republic of Korea.
@article{CRMATH_2021__359_2_113_0,
     author = {Cho, Yunhyung},
     title = {Remark on the {Betti} numbers for {Hamiltonian} circle actions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {113--117},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {2},
     year = {2021},
     doi = {10.5802/crmath.127},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.127/}
}
TY  - JOUR
AU  - Cho, Yunhyung
TI  - Remark on the Betti numbers for Hamiltonian circle actions
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 113
EP  - 117
VL  - 359
IS  - 2
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.127/
DO  - 10.5802/crmath.127
LA  - en
ID  - CRMATH_2021__359_2_113_0
ER  - 
%0 Journal Article
%A Cho, Yunhyung
%T Remark on the Betti numbers for Hamiltonian circle actions
%J Comptes Rendus. Mathématique
%D 2021
%P 113-117
%V 359
%N 2
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.127/
%R 10.5802/crmath.127
%G en
%F CRMATH_2021__359_2_113_0
Cho, Yunhyung. Remark on the Betti numbers for Hamiltonian circle actions. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 113-117. doi : 10.5802/crmath.127. http://www.numdam.org/articles/10.5802/crmath.127/

[1] Atiyah, Michael F.; Bott, Raoul The moment map and equivariant cohomology, Topology, Volume 23 (1984) no. 1, pp. 1-28 | DOI | MR | Zbl

[2] Audin, Michèle Topology of Torus actions on symplectic manifolds Second revised edition, Progress in Mathematics, 93, Birkhäuser, 2004 | Zbl

[3] Berline, Nicole; Vergne, M. Classes charactéristiques équivariantes. Formule de localisation en cohomologie équivariante, C. R. Math. Acad. Sci. Paris, Volume 295 (1982), pp. 539-541 | Zbl

[4] Cho, Yunhyung Unimodality of Betti numbers for Hamiltonian circle actions with index-increasing moment maps, Int. J. Math., Volume 27 (2016) no. 5, 1650043, 14 pages | MR | Zbl

[5] Cho, Yunhyung Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions I, Int. J. Math., Volume 30 (2019) no. 6, 1950032, 71 pages | MR | Zbl

[6] Cho, Yunhyung Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions II (2021) (https://arxiv.org/abs/1904.10962, to appear in International Journal of Mathematics)

[7] Cho, Yunhyung Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions III (2021) (https://arxiv.org/abs/1905.07292, to appear in International Journal of Mathematics) | MR

[8] Cho, Yunhyung; Kim, Min Kyu Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points, Math. Res. Lett., Volume 21 (2014) no. 4, pp. 691-696 | MR | Zbl

[9] Delzant, Thomas Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. Fr., Volume 116 (1988) no. 3, pp. 315-339 | DOI | Zbl

[10] Goertsches, Oliver; Konstantis, Panagiotis; Zoller, Leopold GKM theory and Hamiltonian non-Kähler actions in dimension 6, Adv. Math., Volume 368 (2020), 107141 | Zbl

[11] Goldin, Rebecca F.; Tolman, Susan Towards Generalizing Schubert Calculus in the Symplectic Category, J. Symplectic Geom., Volume 7 (2009) no. 4, pp. 449-473 | DOI | MR | Zbl

[12] Jeffrey, Lisa; Holm, Tara; Karshon, Yael; Lerman, Eugene M.; Meinrenken, Eckhard Moment maps in various geometries, 2005 (available at http://www.birs.ca/workshops/2005/05w5072/report05w5072.pdf)

[13] Karshon, Yael Periodic Hamiltonian flows on four-dimensional manifolds, Memoirs of the American Mathematical Society, 672, American Mathematical Society, 1999 | MR | Zbl

[14] Kirwan, Frances Clare Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, 31, Princeton University Press, 1984 | MR | Zbl

[15] McDuff, Dusa Some 6-dimensional Hamiltonian S 1 -manifolds, J. Topol., Volume 2 (2009) no. 3, pp. 589-623 | DOI | MR | Zbl

[16] McDuff, Dusa; Tolman, Susan Topological properties of Hamiltonian circle actions, Int. Math. Res. Pap., Volume 2006 (2006) no. 4, 72826 | Zbl

[17] Tolman, Susan Examples of non-Kähler Hamiltonian torus actions, Invent. Math., Volume 131 (1998) no. 2, pp. 299-310 | DOI | MR | Zbl

[18] Tolman, Susan On a symplectic generalization of Petrie’s conjecture, Trans. Am. Math. Soc., Volume 362 (2010) no. 8, pp. 3963-3996 | DOI | MR | Zbl

[19] Woodward, Chris Multiplicity-free Hamiltonian actions need not be Kähler, Invent. Math., Volume 131 (1998) no. 2, pp. 311-319 | DOI | MR | Zbl

Cited by Sources: