Analytic continuation of a parametric polytope and wall-crossing

Details Analytic continuation of a parametric polytope and wall-crossing Analytic continuation of a parametric polytope and wall-crossing

2011-04-11

arxiv.org/abs/1104.1885v1

Mathematics

Algebraic Geometry

61 pages, 17 figures

Scientific paper

We define a set theoretic "analytic continuation" of a polytope defined by inequalities. For the regular values of the parameter, our construction coincides with the parallel transport of polytopes in a mirage introduced by Varchenko. We determine the set-theoretic variation when crossing a wall in the parameter space, and we relate this variation to Paradan's wall-crossing formulas for integrals and discrete sums. As another application, we refine the theorem of Brion on generating functions of polytopes and their cones at vertices. We describe the relation of this work with the equivariant index of a line bundle over a toric variety and Morelli constructible support function.

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