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

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, lambda), 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 (Q, lambda) or on Sigma 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.