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

We examine the integral cohomology rings of certain families of $2 n$ –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.

Keywords

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

Mathematical Subject Classification 2010

Primary: 14M25, 55N91, 57R18

Secondary: 13F55, 52B11

References

Publication

Received: 20 December 2016

Revised: 22 March 2017

Accepted: 4 April 2017

Published: 4 October 2017

Authors

Anthony Bahri
Department of Mathematics Rider University Lawrenceville, NJ United States

Soumen Sarkar
Department of Mathematics Indian Institute of Technology Madras Chennai India

Jongbaek Song
Department of Mathematical Sciences KAIST Daejeon South Korea

Volume 17, issue 6 (2017)

Anthony Bahri, Soumen Sarkar and Jongbaek Song

Abstract

Keywords

Mathematical Subject Classification 2010

References

Publication

Authors