Integral cohomology ring of toric orbifolds

Anthony Bahri, Soumen Sarkar, Jongbaek Song

Published 2015-09-10Version 1

We study the integral cohomology rings of certain families of $2n$- dimensional orbifolds $X$ that are equipped with a well-behaved action of the $n$-dimensional torus. The orbifolds arise from two distinct but closely related combinatorial sources: from characteristic pairs $(Q, {\lambda})$, where $Q$ is a simple convex $n$-polytope and ${\lambda}$ a labelling of its facets, and from $n$-dimensional fans $\Sigma$. In recent literature, they are referred to as toric orbifolds and singular toric varieties respectively. Our first main result provides conditions on $(Q, {\lambda})$ or on $\Sigma$ which ensure that the integral cohomology groups $H^\ast(X)$ of the associated orbifolds are concentrated in even degrees. Our second main result assumes these condition to be true, and expresses the graded ring $H^\ast(X)$ as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. We illustrate our ideas with a family of examples that involve orbifold towers, including 2-stage orbifold Hirzebruch surfaces whose integral cohomology rings may be presented in an attractive form. To conclude, we discuss the effects of the simplicial wedge construction (or $J$-construction, in the orbifold context) on our combinatorial conditions and cohomology calculations.