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Equivariant \(\varvec{K}\)-theory of quasitoric manifolds

  • Jyoti Dasgupta
  • Bivas Khan
  • V UmaEmail author
Article
  • 47 Downloads

Abstract

Let \(X(Q,\Lambda )\) be a quasitoric manifold associated to a simple convex polytope Q and characteristic function \(\Lambda \). Let \(T\cong ({S}^1)^n\) denote the compact n-torus acting on \(X=X(Q,\Lambda )\). The main aim of this article is to give a presentation of the T-equivariant K-ring of X, as a Stanley–Reisner ring over \(K^*(pt)\). We also derive the presentation for the ordinary K-ring of X.

Keywords

Quasitoric manifolds K-theoretic face ring simple convex polytope 

1991 Mathematics Subject Classification

55N15 19L47 

Notes

Acknowledgement

The first and the second authors thank Council of Scientific and Industrial Research (CSIR) for their financial support.

References

  1. 1.
    Baggio S, Equivariant \(K\)-theory of smooth toric varieties, Tohoku Math. J. Second Ser. 59(2) (2007) 203–231MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bredon Glen E, Introduction to compact transformation groups, (1972) (New York–London: Academic Press) Pure and Applied Mathematics, vol. 46Google Scholar
  3. 3.
    Buchstaber Victor M and Panov Taras E, Toric topology, Mathematical Surveys and Monographs 204 (2015) (American Mathematical Society, Providence, RI)Google Scholar
  4. 4.
    Dasgupta J, Khan B and Uma V, Cohomology of torus manifold bundles, Math. Slovaca 69(3) (2019) 685–698,  https://doi.org/10.1515/ms-2017-0257 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Davis Michael W and Januszkiewicz T, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62(2) (1991) 417–451MathSciNetCrossRefGoogle Scholar
  6. 6.
    Harada M, Henriques A and Holm Tara S, Computation of generalized equivariant cohomologies of Kac-Moody flag varieties, Adv. Math. 197(1) (2005) 198–221MathSciNetCrossRefGoogle Scholar
  7. 7.
    Harada M, Holm Tara S, Ray N and Williams G, The equivariant K-theory and cobordism rings of divisive weighted projective spaces, Tohoku Math. J. (2) 68(4) (2016) 487–513MathSciNetCrossRefGoogle Scholar
  8. 8.
    Harada M and Landweber Gregory D, Surjectivity for Hamiltonian \(G\)-spaces in \(K\)-theory, Trans. Am. Math. Soc. 359(12) (2007) 6001–6025MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mukherjee G, Transformation groups, (2005) (New Delhi: Hindustan Book Agency), Symplectic torus actions and toric manifolds, with contributions by C Allday, M Masuda and P SankaranGoogle Scholar
  10. 10.
    Sankaran P and Uma V, \(K\)-theory of quasi-toric manifolds, Osaka J. Math. 44(1) (2007) 71–89MathSciNetzbMATHGoogle Scholar
  11. 11.
    Sarkar S and Uma V, Equivariant \(K\)-theory and cobordism rings of divisive toric varieties and toric orbifolds (2018) arXiv:1804.07883 [math.AG]
  12. 12.
    Segal G, Equivariant \(K\)-theory, Institut des Hautes Études Scientifiques, Publ. Mathématiques 34 (1968) 129–151CrossRefGoogle Scholar
  13. 13.
    Vezzosi G and Vistoli A, Higher algebraic \(K\)-theory for actions of diagonalizable groups, Inventiones Mathematicae 153(1) (2003) 1–44MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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