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Convex polytopes, Coxeter orbifolds and torus actions. (English) Zbl 0733.52006
This is a comprehensive survey with remarkable new results on the topic indicated in the title. The authors start with a convex n-polytope \(P^ n\) that is called simple, if the number of codimension-one faces meeting at each vertex is n. They consider two types of group actions \(Z^ n_ 2\) resp. \(T^ n\) (torus action) on n-manifold \(M^ n\) resp. 2n-manifold \(M^{2n}\), such that the orbit space is just \(P^ n\cong M^ n/Z^ n_ 2\) resp. \(\cong M^{2n}/T^ n\). Up to an automorphism, the group action is required to be locally isomorphic to the standard representation of \(Z^ n_ 2\) on \({\mathbb{R}}^ 2\) resp. \(T^ n\) on \({\mathbb{C}}^ n\). In the first case \(M^ n\) is called a “small cover” of \(P^ n\), in the second \(M^{2n}\) is a “toric manifold” over \(P^ n.\)
If m is the number of codimension-one faces of \(P^ n\), then a homomorphism \(\lambda\) : \(Z^ m_ 2\to Z^ n_ 2\) resp. \(Z^ m\to Z^ n\) specifies an isotropy subgroup for each codimension-one face. \(\lambda\) is called “characteristic function” for \(M^ n\) resp. \(M^{2n}\). Let \(f_ i\) denote the number of i-faces of \(P^ n\), and let \(h_ j\) denote the coefficient of \(t^{n-j}\) in \(\sum f_ i(t- 1)^ i.\) Then the f-vector \((f_ 0,...,f_ n)\) and the h-vector \((h_ 0,...,h_ n)\) determine one another. P. McMullen proved the inequality \(h_ i\leq \left( \begin{matrix} m-n+i-1\\ i\end{matrix} \right)\) and conjectured simple combinatorial conditions on a sequence of integers to be the h- vector of a simple convex n-polytope. The sufficiency of these conditions was proved by Billera and Lee and necessity by Stanley [see A. Brøndsted, An introduction to convex polytopes, Springer, New-York (1983; Zbl 0509.52001)]. Such problems, particularly that of R. Stanley [Stud. Appl. Math. 54, 135-142 (1975; Zbl 0308.52009)] have led to injections of research aspects as follows
(A) The choice of a generic vector (never tangent to a proper face of \(P^ n)\) allows to attach an integer-valued index to each vertex so that the number of vertices of index i is \(h_ i\). This can be used to define here a cell structure which is perfect in the sense of Morse theory [see also A. G. Khovanskii, Funct. Anal. Appl. 20, 41-50 (1986); translation from Funkts. Anal. Prilozh. 20, No.1, 50-61 (1986; Zbl 0597.51014)].
(B) The “face ring“ of the simplicial complex K, dual to the boundary complex of \(P^ n\), is characterized as a “Cohen-Macaulay complex” in G. Reisner, Adv. Math. 21, 30-49 (1976; Zbl 0345.13017). This topic is extensively studied in Sections 4 and 5 of the paper.
(C) Stanley established the necessity of McMullen’s condition by studying a certain quasi-smooth projective variety associated to \(P^ n\), called a “toric variety”. Then the existence of a Kähler class yields the result. Nonsingular toric varieties are toric manifolds in the sense of this paper, however, the converse does not hold. In Section 6 the authors consider the tangent bundle of small covers and of toric manifolds. It is stably isomorphic to a sum of real resp. complex line bundles, and this leads to formulae for its characteristic classes.

MSC:
52B70 Polyhedral manifolds
57Q91 Equivariant PL-topology
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
14L30 Group actions on varieties or schemes (quotients)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
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