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

Abstract

We examine the integral cohomology rings of certain families of [math] –dimensional orbifolds [math] that are equipped with a well-behaved action of the [math] –dimensional real torus. These orbifolds arise from two distinct but closely related combinatorial sources, namely from characteristic pairs [math] , where [math] is a simple convex [math] –polytope and [math] a labeling of its facets, and from [math] –dimensional fans [math] . In the literature, they are referred as toric orbifolds and singular toric varieties, respectively. Our first main result provides combinatorial conditions on [math] or on [math] which ensure that the integral cohomology groups [math] of the associated orbifolds are concentrated in even degrees. Our second main result assumes these conditions to be true, and expresses the graded ring [math] as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. Also, we compute several examples.