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 (Q, λ), where Q is a simple convex n-polytope and λ a labelling of its facets, and from ndimensional 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 condition 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.

Generic torus orbit closures in Schubert varieties

“…The aim of this section is to study the Poincaré polynomial of certain toric varieties. The main idea is to use a retraction sequence of a polytope introduced in [2,3,16]. We first begin by reviewing some facts on projective toric varieties associated with lattice polytopes and their properties which are necessary to our aim.…”

“…Remark 2.3. Every simple polytope has at least one retraction sequence (see [3,Proposition 2.3]). However, some singular polytopes, for instance, the rhombododecahedron appeared in [15] do not admit a retraction sequence.…”

“…Indeed, there exist two combinatorially equivalent lattice polytopes for which the associated toric varieties have different Betti numbers, see [15]. There are several approaches to computing the Betti numbers of singular toric varieties, e.g., using spectral sequences (see [9,12] and [8, §12.3]), Bia lynicki-Birula decomposition [4], retraction sequences on polytopes [2,3,16].…”

Torus Orbifolds with Two Fixed Points

“…which is called an orbifold lens space in [6]. We note that this orbifold L(Λ) has a natural T n -action.…”

“…For toric manifolds the ring of piecewise polynomials on the fan is isomorphic to the face-ring of its quotient but this is not true for orbifolds in general. In [6] it is shown that the equivariant cohomology of projective toric orbifolds, under the condition of vanishing odd degree cohomology, can be realized as a subring of the usual face-ring of the fan that satisfies an intergrality condition.…”