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Intersection cohomology on nonrational polytopes. (English) Zbl 1024.52005
The first proof of R. P. Stanley [Adv. Math. 35, 236–238 (1980; Zbl 0427.52006)] of the necessity part of the reviewer’s \(g\)-conjecture for simplicial polytopes used fairly high-powered algebraic geometry (of toric varieties). Subsequently, the reviewer [Invent. Math. 113, No. 2, 419–444 (1993; Zbl 0803.52007)] (with simplifications in [Discrete Comput. Geom. 15, No. 4, 363–388 (1996; Zbl 0849.52011)]) found a proof which works entirely within convexity. In [Adv. Stud. Pure Math. 11, 187–213 (1987; Zbl 0652.52007)] R. P. Stanley also defined a generalized \(h\)-vector for all polytopes, and showed (again using toric varieties) that, for rational polytopes, its terms are monotone. However, in general, polytopes will not be rational, even up to combinatorial type, and so the search has been on for some time to discover techniques which can replace those coming from algebraic geometry. The foundations for such a technique are described in the present paper. Specifically, the authors have devised an intersection cohomology for normal fans of arbitrary convex polytopes. It would be infeasible to go into the somewhat complicated details here; suffice it to say that they have also identified the corresponding Lefschetz element, which they suggest should yield Stanley’s conjecture.
Two further comments are appropriate. First, much the same theory has been constructed independently by G. Barthel, J.-P. Brasselet, K.-H. Fieseler and L. Kaup [TĂ´hoku Math. J., II. Ser. 54, No. 1, 1–41 (2002; Zbl 1055.14024)]. Second, Kalle Karu (as yet unpublished) has built on these structures to complete the proof of Stanley’s conjecture.

MSC:
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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