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On the integral cohomology ring of toric orbifolds and singular toric varieties. (English) Zbl 1386.14187
This works develops strategies that help in the calculation of the integral cohomology of some families of \(2n\)-dimensional orbifolds equipped with an action of the \(n\)-dimensional real torus: toric orbifolds (those orbifolds arising from characteristic pairs \((Q,\lambda)\), where \(Q\) is a simple convex \(n\)-polytope and \(\lambda\) is a labeling of its codimension-\(1\) faces), and singular toric varieties (those orbifolds arising from \(n\)-dimensional fans \(\Sigma\).) The construction of the objects in either of these families is combinatorial in nature, and this is reflected directly in the results obtained by the authors, and in the methods needed to prove those results.
A topological space is called even if its integral cohomology is torsion-free and is concentrated in even degrees, and for such spaces one can for example directly deduce the \(K\)-theory or complex cobordism groups. The first part of this paper finds conditions for the evenness of toric orbifolds. This is Theorem 1.1, which relies on the notion of retraction sequences for a simple polytope \(Q\), a concept explored at length in Section 3: these are carefully chosen sequences of polytopal complexes whose initial term is \(Q\) and whose final term is a vertex of \(Q\). For each polytopal complex \(B\) appearing in a retraction sequence, one can use \(\lambda\) to define a collection of finite groups, each one associated with a so-called free vertex of \(B\). The (combinatorial) condition in Theorem 1.1 that guarantees the evenness of the toric orbifold has to do with the order of such groups for all possible polytopal complexes \(B\) in all possible retraction sequences for \(Q\).
The second part of the paper deals with some projective toric orbifolds, which are toric varieties \(X_\Sigma\) encoded by a fan \(\Sigma\) in \({\mathbb R}^n\) and with an action by the \(n\)-dimensional real torus. If \(X_\Sigma\) is smooth, its cohomology is a quotient of the Stanley-Reisner ring of the fan \(\Sigma\) by some linear relations. For the toric orbifold corresponding to \(\Sigma\), this result will only be true over the rationals. To circumvent this problem, and to be able to obtain similar results for the cohomology of singular toric varieties, the authors define here the weighted Stanley-Reisner ring for a fan, which is a subring of the above Stanley-Reisner ring consisting of polynomials satisfying the integrality condition in definition 5.2. The main result in this second part is then Theorem 5.3, which declares that the integral cohomology of a projective toric orbifold \(X_\Sigma\) (which is assumed to be even) is a quotient of the weighted Stanley-Reisner ring by linear relations similar to those that appeared in the previous result for the smooth case.
The direct combination of Theorems 1.1 and 5.3 is Theorem 1.2: given a polytopal fan \(\Sigma\) in \({\mathbb R}^n\) whose corresponding characteristic pair \((Q,\lambda)\) satisfies the condition of Theorem 1.1, the associated projective toric variety \(X_\Sigma\) has integral cohomology given by the quotient in the text of Theorem 5.3. The paper ends with the calculation of the integral cohomology of a projective toric orbifold that is not a weighted projective space, thus demonstrating the usefulness of the developed generalizations in solving further, previously untackled cases.

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
55N91 Equivariant homology and cohomology in algebraic topology
57R18 Topology and geometry of orbifolds
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
52B11 \(n\)-dimensional polytopes
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