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 $\left(Q,\lambda \right)$, where $Q$ is a simple convex $n$–polytope and $\lambda$ a labeling of its facets, and from $n$–dimensional fans $\Sigma$. In the literature, they are referred as toric orbifolds and singular toric varieties, respectively. Our first main result provides combinatorial conditions on $\left(Q,\lambda \right)$ or on $\Sigma$ which ensure that the integral cohomology groups $H^{\ast }\left(X\right)$ of the associated orbifolds are concentrated in even degrees. Our second main result assumes these conditions to be true, and expresses the graded ring $H^{\ast }\left(X\right)$ as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. Also, we compute several examples.