Algebraic & Geometric Topology

On the integral cohomology ring of toric orbifolds and singular toric varieties

Anthony Bahri, Soumen Sarkar, and Jongbaek Song

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We examine the integral cohomology rings of certain families of 2n–dimensional orbifolds X that are equipped with a well-behaved action of the n–dimensional real torus. These orbifolds arise from two distinct but closely related combinatorial sources, namely from characteristic pairs (Q,λ), where Q is a simple convex n–polytope and λ a labeling of its facets, and from n–dimensional fans Σ. In the literature, they are referred as toric orbifolds and singular toric varieties, respectively. Our first main result provides combinatorial conditions on (Q,λ) or on Σ which ensure that the integral cohomology groups H(X) of the associated orbifolds are concentrated in even degrees. Our second main result assumes these conditions to be true, and expresses the graded ring H(X) as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. Also, we compute several examples.

Article information

Algebr. Geom. Topol., Volume 17, Number 6 (2017), 3779-3810.

Received: 20 December 2016
Revised: 22 March 2017
Accepted: 4 April 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 55N91: Equivariant homology and cohomology [See also 19L47] 57R18: Topology and geometry of orbifolds
Secondary: 13F55: Stanley-Reisner face rings; simplicial complexes [See also 55U10] 52B11: $n$-dimensional polytopes

toric orbifold quasitoric orbifold toric variety lens space equivariant cohomology Stanley–Reisner ring piecewise polynomial


Bahri, Anthony; Sarkar, Soumen; Song, Jongbaek. On the integral cohomology ring of toric orbifolds and singular toric varieties. Algebr. Geom. Topol. 17 (2017), no. 6, 3779--3810. doi:10.2140/agt.2017.17.3779.

Export citation


  • A Bahri, M Franz, N Ray, The equivariant cohomology ring of weighted projective space, Math. Proc. Cambridge Philos. Soc. 146 (2009) 395–405
  • A Bahri, S Sarkar, J Song, Shellability and retraction sequences, in preparation
  • A Borel, Seminar on transformation groups, Annals of Mathematics Studies 46, Princeton Univ. Press (1960)
  • V M Buchstaber, T E Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series 24, Amer. Math. Soc., Providence, RI (2002)
  • W Buczynska, Fake weighted projective spaces, preprint (2008)
  • D A Cox, J B Little, H K Schenck, Toric varieties, Graduate Studies in Mathematics 124, Amer. Math. Soc., Providence, RI (2011)
  • V I Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978) 85–134 In Russian; translated in Russian Math. Surveys 33 (1978) 97–154
  • A Darby, S Kuroki, J Song, The equivariant cohomology of torus orbifolds, in preparation
  • M W Davis, T Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) 417–451
  • S Fischli, On toric varieties, PhD thesis, Universität Bern (1992) Available at \setbox0\makeatletter\@url {\unhbox0
  • M Franz, V Puppe, Exact cohomology sequences with integral coefficients for torus actions, Transform. Groups 12 (2007) 65–76
  • W Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton Univ. Press (1993)
  • M Harada, A Henriques, T S Holm, Computation of generalized equivariant cohomologies of Kac–Moody flag varieties, Adv. Math. 197 (2005) 198–221
  • M Harada, T S Holm, N Ray, G Williams, The equivariant $K$–theory and cobordism rings of divisive weighted projective spaces, Tohoku Math. J. 68 (2016) 487–513
  • A Jordan, Homology and cohomology of toric varieties, PhD thesis, University of Konstanz (1998) Available at \setbox0\makeatletter\@url {\unhbox0
  • J Jurkiewicz, Torus embeddings, polyhedra, $k^\ast$–actions and homology, Dissertationes Math. $($Rozprawy Mat.$)$ 236, Polska Akademia Nauk. Instytut Matematyczny, Warsaw (1985)
  • A M Kasprzyk, Bounds on fake weighted projective space, Kodai Math. J. 32 (2009) 197–208
  • T Kawasaki, Cohomology of twisted projective spaces and lens complexes, Math. Ann. 206 (1973) 243–248
  • H Kuwata, M Masuda, H Zeng, Torsion in the cohomology of torus orbifolds, preprint (2016)
  • M Poddar, S Sarkar, On quasitoric orbifolds, Osaka J. Math. 47 (2010) 1055–1076
  • S Sarkar, D Y Suh, A new construction of lens spaces, preprint (2013)
  • S Sarkar, V Uma, Equivariant k–theory and cobordism rings of toric orbifolds, in preparation
  • G M Ziegler, Lectures on polytopes, Graduate Texts in Mathematics 152, Springer (1995)