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Analytic continuation of a parametric polytope and wall-crossing. (English) Zbl 1303.52003

Björner, A. et al., Configuration spaces. Geometry, combinatorics and topology. Pisa: Edizioni della Normale (ISBN 978-88-7642-430-4/pbk; 978-88-7642-431-1/ebook). Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series 14, 111-172 (2012).
The authors give a comprehensive presentation of topological and combinatorial properties of parametric polytopes. An important role play integrals and discrete sums of polynomials over polytopes. A certain decomposition of the involved polytopes as signed sums of cones leads to the Brianchon-Gram decomposition. The focus of this paper lies on various aspects of “wall-crossing”: combinatorial wall-crossing, geometric wall-crossing, polynomiality and wall-crossing for integrals and sums, Paradan’s convolution wall-crossing formulas. As an application, the theorem of Brion on generating functions of polytopes and their cones (at vertices) is refined. The paper concludes with relations to the cohomology of line bundles over toric varieties.
For the entire collection see [Zbl 1253.00010].

MSC:

52B11 \(n\)-dimensional polytopes
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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