Hostname: page-component-857557d7f7-bkbbk Total loading time: 0 Render date: 2025-12-08T03:51:39.594Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivariant cohomology of torus orbifolds

Part of: Differential topology Arithmetic rings and other special rings Homology and cohomology theories Special varieties

Published online by Cambridge University Press:  20 November 2020

Alastair Darby ,
Shintarô Kuroki  and
Jongbaek Song
Alastair Darby
Affiliation:
Department of Pure Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou, China e-mail: Alastair.Darby@xjtlu.edu.cn
Shintarô Kuroki
Affiliation:
Department of Applied Mathematics, Okayama University of Science, Okayama, Japan email: kuroki@xmath.ous.ac.jp
Jongbaek Song*
Affiliation:
School of Mathematics, KIAS, Seoul, Republic of Korea
*
e-mail: jongbaek.song@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We calculate the integral equivariant cohomology, in terms of generators and relations, of locally standard torus orbifolds whose odd degree ordinary cohomology vanishes. We begin by studying GKM-orbifolds, which are more general, before specializing to half-dimensional torus actions.

Keywords

GKM-orbifoldtorus orbifoldequivariant cohomologyGKM-theoryface ring

MSC classification

Primary: 14M25: Toric varieties, Newton polyhedra
Secondary: 55N91: Equivariant homology and cohomology 57R18: Topology and geometry of orbifolds 13F55: Stanley-Reisner face rings; simplicial complexes

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 2 , April 2022 , pp. 299 - 328
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

A.D. was supported by XJTLU RDF-17-02-50. S.K. was supported by JSPS KAKENHI Grant Number 17K14196. J.S. has been supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07048480) and a KIAS Individual Grant (MG076101) at Korea Institute for Advanced Study.

References

Abe, H. and Matsumura, T., Equivariant cohomology of weighted Grassmannians and weighted Schubert classes. Int. Math. Res. Not. 2015(2015), no. 9, 2499–2524.CrossRefGoogle Scholar
Bahri, A., Franz, M., and Ray, N., The equivariant cohomology ring of weighted projective space . Math. Proc. Camb. Philos. Soc. 146(2009), no. 2, 395–405.CrossRefGoogle Scholar
Blume, M., Toric orbifolds associated to Cartan matrices . Ann. Inst. Fourier (Grenoble) 65(2015), no. 2, 863–901.CrossRefGoogle Scholar
Bahri, A., Notbohm, D., Sarkar, S., and Song, J., On integral cohomology of certain orbifolds , Int. Math. Res. Not. (2019), rny283. https://doi.org/10.1093/imrn/rny283 Google Scholar
Buchstaber, V. M. and Panov, T. E., Toric topology . In: Mathematical Surveys and Monographs, Vol. 204, American Mathematical Society, Providence, RI, 2015.Google Scholar
Bahri, A., Sarkar, S., and Song, J., On the integral cohomology ring of toric orbifolds and singular toric varieties . Algebr. Geom. Topol. 17(2017), no. 6, 3779–3810.CrossRefGoogle Scholar
Cox, D. A., Little, J. B., and Schenck, H. K., Toric varieties . In: Graduate Studies in Mathematics, Vol. 124, American Mathematical Society, Providence, RI, 2011.Google Scholar
Corti, A. and Reid, M., Weighted Grassmannians algebraic geometry . De Gruyter, Berlin, Germany, 2002, pp. 141–163.Google Scholar
Davis, M. W. and Januszkiewicz, T., Convex polytopes, Coxeter orbifolds and torus actions . Duke Math. J. 62(1991), no. 2, 417–451.CrossRefGoogle Scholar
Darby, A., Kuroki, S., and Song, J., Torus orbifolds with two fixed points . In: Algebraic topology and related topics, Trends Math., Birkhäuser, Springer, Singapore, 2019, pp. 67–85.CrossRefGoogle Scholar
Franz, M. and Puppe, V., Exact cohomology sequences with integral coefficients for torus actions . Transform. Groups 12(2007), no. 1, 65–76.CrossRefGoogle Scholar
Franz, M., Describing toric varieties and their equivariant cohomology . Colloq. Math. 121(2010), no. 1, 1–16.CrossRefGoogle Scholar
Fulton, W., Introduction to toric varieties . In: Annals of Mathematics Studies, The William H. Roever Lectures in Geometry, 131, Princeton University Press, Princeton, NJ, 1993.Google Scholar
Galaz-García, F., Kerin, M., Radeschi, M., and Wiemeler, M., Torus orbifolds, slice-maximal torus actions, and rational ellipticity . Int. Math. Res. Not. (2018), no. 18, 5786–5822.CrossRefGoogle Scholar
Goresky, M., Kottwitz, R., and MacPherson, R., Equivariant cohomology, Koszul duality, and the localization theorem . Invent. Math. 131(1998), no. 1, 25–83.CrossRefGoogle Scholar
Guillemin, V. and Zara, C., 1-Skeleta, Betti numbers, and equivariant cohomology . Duke Math. J. 107(2001), no. 2, 283–349.CrossRefGoogle Scholar
Hattori, A. and Masuda, M., Theory of multi-fans . Osaka J. Math. 40(2003), no. 1, 1–68.Google Scholar
Kawasaki, T., Cohomology of twisted projective spaces and lens complexes . Math. Ann. 206(1973), 243–248.CrossRefGoogle Scholar
Kuwata, H., Masuda, M., and Zeng, H., Torsion in the cohomology of torus orbifolds . Chin. Ann. Math. Ser. B 38(2017), no. 6, 1247–1268.CrossRefGoogle Scholar
Lerman, E. and Tolman, S., Hamiltonian torus actions on symplectic orbifolds and toric varieties . Trans. Am. Math. Soc. 349(1997), no. 10, 4201–4230.CrossRefGoogle Scholar
Maeda, H., Masuda, M., and Panov, T., Torus graphs and simplicial posets . Adv. Math. 212(2007), no. 2, 458–483.CrossRefGoogle Scholar
Masuda, M. and Panov, T., On the cohomology of torus manifolds . Osaka J. Math. 43(2006), no. 3, 711–746.Google Scholar
Novoseltsev, A. Y., Toric lattices framework for Sage. SageMath, 2010. http://doc.sagemath.org/html/en/reference/discrete_geometry/sage/geometry/toric_lattice.html Google Scholar
Poddar, M. and Sarkar, S., On quasitoric orbifolds . Osaka J. Math. 47(2010), no. 4, 1055–1076.Google Scholar
Qureshi, M. I. and Szendröi, B., Constructing projective varieties in weighted flag varieties . Bull. Lond. Math. Soc. 43(2011), no. 4, 786–798.CrossRefGoogle Scholar
Stein, W. A., Sage mathematics software. SageMath, 2007. http://www.sagemath.org Google Scholar
Satake, I., On a generalization of the notion of manifold . Proc. Natl. Acad. Sci. 42(1956), 359–363.CrossRefGoogle Scholar