On the Integral Cohomology Ring of Toric Orbifolds and Singular Toric Varieties · arXivDesk
1509.03228Sep 10, 201525 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
On the Integral Cohomology Ring of Toric Orbifolds and Singular Toric Varieties
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
Nearby in the stack
n
-polytope and
λ
a labelling of its facets, and from
n
-dimensional 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.