On the Integral Cohomology Ring of Toric Orbifolds and Singular Toric Varieties

TL;DR

This paper investigates the integral cohomology rings of certain toric orbifolds and singular toric varieties, providing combinatorial conditions for even-degree cohomology and explicit algebraic descriptions.

Contribution

It establishes combinatorial criteria for even-degree cohomology and derives algebraic presentations of cohomology rings for these orbifolds and varieties.

Findings

Cohomology groups are concentrated in even degrees under specific conditions.

Provides algebraic presentations of cohomology rings as polynomial quotients.

Includes computed examples illustrating the theoretical results.

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 $H^*(X)$ as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. Also, we compute several examples.