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.
The CW structure of certain spaces, such as effective orbifolds, can be too complicated for computational purposes. In this paper we use the concept of q-CW complex structure on an orbifold, to detect torsion in its integral cohomology. The main result can be applied to well known classes of orbifolds or algebraic varieties having orbifold singularities, such as toric orbifolds, simplicial toric varieties, torus orbifolds and weighted Grassmannians.
The main objects of this paper are torus orbifolds that have exactly two fixed points. We study the equivariant topological type of these orbifolds and consider when we can use the results of [8] to compute its integral equivariant cohomology, in terms of generators and relations, coming from the corresponding orbifold torus graph.
We calculate the integral equivariant cohomology, in terms of generators and relations, of locally standard torus orbifolds whose odd degree ordinary cohomology vanishes. We begin by studying GKM-orbifolds, which are more general, before specialising to half-dimensional torus actions.
The closure of a generic torus orbit in the flag variety
G
/
B
G/B
of type
A
A
is known to be a permutohedral variety, and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in
G
/
B
G/B
. When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.
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