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 $(Q,\lambda )$, where $Q$ is a simple convex $n$-polytope and $\lambda$ a labelling 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 $H^{\ast }(X)$ of the associated orbifolds are concentrated in even degrees. Our second main result assumes these condition to be true, and expresses the graded ring $H^{\ast }(X)$ as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. Also, we compute several examples.

25 pages. The title has beed changed. The assumption of retraction sequence becomes weaker than the previous version, hence the main theorem covers wider class of orbifolds. Section 6 of the previous version has been separated into an independent article now in preparation. Some material has been added to improve the exposition

References in corpus (3)

• Fake weighted projective spaces
• The equivariant cohomology of weighted projective space
• Torsion in the cohomology of torus orbifolds

Cited by in corpus (9)

• Generic torus orbit closures in Schubert varieties
• Torus orbifolds with two fixed points
• Lickorish type construction of manifolds over simple polytopes
• Equivariant cohomology of torus orbifolds
• GKM theory for orbifold stratified spaces and application to singular toric varieties
• Equivariant K-theory of toric orbifolds
• The homotopy classification of four-dimensional toric orbifolds
• On integral cohomology of certain orbifolds
• Poincare polynomials of generic torus orbit closures in Schubert varieties