Analytic continuation of a parametric polytope and wall-crossing

TL;DR

This paper introduces a set-theoretic analytic continuation for polytopes defined by inequalities, analyzing how these polytopes change when parameters cross walls, and relating this to existing wall-crossing formulas and toric geometry.

Contribution

It defines a new set-theoretic analytic continuation of polytopes, relates it to wall-crossing phenomena, and refines existing theorems on generating functions and toric varieties.

Findings

Set-theoretic variation across walls is explicitly characterized.

Relation established between the continuation and Paradan's wall-crossing formulas.

Refinement of Brion's theorem on generating functions of polytopes.

Abstract

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.