Almost complex torus manifolds - a problem of Petrie type
HTML articles powered by AMS MathViewer
- by Donghoon Jang;
- Proc. Amer. Math. Soc. 152 (2024), 3153-3164
- DOI: https://doi.org/10.1090/proc/16768
- Published electronically: May 15, 2024
- HTML | PDF | Request permission
Abstract:
The Petrie conjecture asserts that if a homotopy $\mathbb {CP}^n$ admits a non-trivial circle action, its Pontryagin class agrees with that of $\mathbb {CP}^n$. Petrie proved this conjecture in the case where the manifold admits a $T^n$-action. An almost complex torus manifold is a $2n$-dimensional compact connected almost complex manifold equipped with an effective $T^n$-action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a $2n$-dimensional almost complex torus manifold $M$ only shares the Euler number with the complex projective space $\mathbb {CP}^n$, the graph of $M$ agrees with the graph of a linear $T^n$-action on $\mathbb {CP}^n$. Consequently, $M$ has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch $\chi _y$-genus, Todd genus, and signature as $\mathbb {CP}^n$, endowed with the standard linear action. Furthermore, if $M$ is equivariantly formal, the equivariant cohomology and the Chern classes of $M$ and $\mathbb {CP}^n$ also agree.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
- Nicole Berline and Michèle Vergne, Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 9, 539–541 (French, with English summary). MR 685019
- Anand Dessai and Burkhard Wilking, Torus actions on homotopy complex projective spaces, Math. Z. 247 (2004), no. 3, 505–511. MR 2114425, DOI 10.1007/s00209-003-0614-z
- Alastair Darby, Torus manifolds in equivariant complex bordism, Topology Appl. 189 (2015), 31–64. MR 3342571, DOI 10.1016/j.topol.2015.03.014
- Italo José Dejter, Smooth $S^1$-manifolds in the homotopy type of $CP^3$, Michigan Math. J. 23 (1976), no. 1, 83–95. MR 402789
- 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
- Oliver Goertsches, Panagiotis Konstantis, and Leopold Zoller, GKM theory and Hamiltonian non-Kähler actions in dimension 6, Adv. Math. 368 (2020), 107141, 17. MR 4088417, DOI 10.1016/j.aim.2020.107141
- Oliver Goertsches and Michael Wiemeler, Positively curved GKM-manifolds, Int. Math. Res. Not. IMRN 22 (2015), 12015–12041. MR 3456711, DOI 10.1093/imrn/rnv046
- Mark Goresky, Robert Kottwitz, and Robert MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25–83. MR 1489894, DOI 10.1007/s002220050197
- Leonor Godinho and Silvia Sabatini, New tools for classifying Hamiltonian circle actions with isolated fixed points, Found. Comput. Math. 14 (2014), no. 4, 791–860. MR 3230015, DOI 10.1007/s10208-014-9204-1
- 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
- Akio Hattori, $\textrm {Spin}^{c}$-structures and $S^{1}$-actions, Invent. Math. 48 (1978), no. 1, 7–31. MR 508087, DOI 10.1007/BF01390060
- David M. James, Smooth $S^1$ actions on homotopy $\textbf {C}\textrm {P}^4$’s, Michigan Math. J. 32 (1985), no. 3, 259–266. MR 803831, DOI 10.1307/mmj/1029003237
- D. Jang and S. Tolman, Hamiltonian circle actions on eight-dimensional manifolds with minimal fixed sets, Transform. Groups 22 (2017), no. 2, 353–359. MR 3649458, DOI 10.1007/s00031-016-9370-0
- Donghoon Jang, Symplectic periodic flows with exactly three equilibrium points, Ergodic Theory Dynam. Systems 34 (2014), no. 6, 1930–1963. MR 3272779, DOI 10.1017/etds.2014.56
- Donghoon Jang, Circle actions on almost complex manifolds with isolated fixed points, J. Geom. Phys. 119 (2017), 187–192. MR 3661531, DOI 10.1016/j.geomphys.2017.05.004
- Donghoon Jang, Circle actions on almost complex manifolds with 4 fixed points, Math. Z. 294 (2020), no. 1-2, 287–319. MR 4050069, DOI 10.1007/s00209-019-02267-z
- Donghoon Jang, Almost complex torus manifolds—graphs and Hirzebruch genera, Int. Math. Res. Not. IMRN 17 (2023), 14594–14609. MR 4637447, DOI 10.1093/imrn/rnac237
- Shoshichi Kobayashi, Fixed points of isometries, Nagoya Math. J. 13 (1958), 63–68. MR 103508, DOI 10.1017/S0027763000023497
- 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
- Mikiya Masuda and Dong Youp Suh, Classification problems of toric manifolds via topology, Toric topology, Contemp. Math., vol. 460, Amer. Math. Soc., Providence, RI, 2008, pp. 273–286. MR 2428362, DOI 10.1090/conm/460/09024
- Mikiya Masuda, On smooth $S^{1}$-actions on cohomology complex projective spaces. The case where the fixed point set consists of four connected components, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 1, 127–167. MR 617869
- Mikiya Masuda, Unitary toric manifolds, multi-fans and equivariant index, Tohoku Math. J. (2) 51 (1999), no. 2, 237–265. MR 1689995, DOI 10.2748/tmj/1178224815
- O. R. Musin, Actions of the circle on homotopy complex projective spaces, Mat. Zametki 28 (1980), no. 1, 139–152, 170 (Russian). MR 585071
- Ted Petrie, Smooth $S^{1}$ actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972), 105–153. MR 296970, DOI 10.1090/S0002-9904-1972-12898-2
- Ted Petrie, Torus actions on homotopy complex projective spaces, Invent. Math. 20 (1973), 139–146. MR 322893, DOI 10.1007/BF01404062
- Etsuo Tsukada and Ryo Washiyama, $S^{1}$-actions on cohomology complex projective spaces with three components of the fixed point sets, Hiroshima Math. J. 9 (1979), no. 1, 41–46. MR 529325
- Susan Tolman, On a symplectic generalization of Petrie’s conjecture, Trans. Amer. Math. Soc. 362 (2010), no. 8, 3963–3996. MR 2638879, DOI 10.1090/S0002-9947-10-04985-8
Bibliographic Information
- Donghoon Jang
- Affiliation: Department of Mathematics, Pusan National University, 2 Busandaehak-ro, 63 Beon-gil, Geumjeong-gu, Busan, Republic of Korea
- MR Author ID: 1083331
- Email: donghoonjang@pusan.ac.kr
- Received by editor(s): August 28, 2022
- Received by editor(s) in revised form: October 30, 2023, and December 22, 2023
- Published electronically: May 15, 2024
- Additional Notes: The author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (2021R1C1C1004158).
- Communicated by: Julie Bergner
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3153-3164
- MSC (2020): Primary 57S12; Secondary 58C30
- DOI: https://doi.org/10.1090/proc/16768
- MathSciNet review: 4753295