Algebraic & Geometric Topology

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

Abstract

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

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

Dates
Revised: 22 March 2017
Accepted: 4 April 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841520

Digital Object Identifier
doi:10.2140/agt.2017.17.3779

Mathematical Reviews number (MathSciNet)
MR3709660

Zentralblatt MATH identifier
06791662

Citation

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. https://projecteuclid.org/euclid.agt/1510841520

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