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.
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References
D. Avis and K. Fukuda, A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra, Discrete Comput. Geom. 8 (1992), no. 3, 295–313, ACM Symposium on Computational Geometry (North Conway, NH, 1991).
V. Baldoni, N. Berline, J. D. Loera, M. Köppe and M. Vergne, Computation of the highest coefficients of weighted Ehrhart quasi-polynomials for a rational polytope, arXiv:1011.1602 [math.CO], 2010.
A. I. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra, In: “New Perspectives in Algebraic Combinatorics”, L. J. Billera, A. Bjorner, C. Greene, R. E. Simion, and R. P. Stanley (eds.), Math. Sci. Res. Inst. Publ., vol. 38, Cambridge Univ. Press, Cambridge, 1999, pp. 91–147.
N. Berline and M. Vergne, Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 9, 539–541.
A. Boysal and M. Vergne, Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 5, 1715–1752.
M. Brion, Points entiers dans les polyèdres convexes, Ann. Sci. École Norm. Sup. 21 (1988), no. 4, 653–663.
M. Brion and M. Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. Reine Angew. Math. 482 (1997), 67–92.
M. Brion and M. Vergne, Lattice points in simple polytopes, J.A.M.S. 10 (1997).
M. Brion and M. Vergne, Residue formulae, vector partition functions and lattice points in rational polytopes, J. Amer. Math. Soc. 10 (1997), 797–833.
R. Cavalieri, P. Johnson and H. Markwig, Chamber structure of double Hurwitz numbers, 2010, arXiv:1003.1805v1.
D. Cox, J. Little and H. Schenck, Toric varieties, 2010, Available at http://www.cs.amherst.edu/dac/toric.html.
W. Dahmen and C. A. Micchelli, The number of solutions to linear Diophantine equations and multivariate splines, Trans. Amer. Math. Soc. 308 (1988), no. 2, 509–532.
C. De Concini, C. Procesi and M. Vergne, Vector partition function and generalized Dahmen-Micchelli spaces, 2008, arXiv:0805.2907v2.
Y. Karshon and S. Tolman, The moment map and line bundles over presymplectic toric manifolds, J. Differential Geom. 38 (1993), no. 3, 465–484.
J. Lawrence, Polytope volume computation, Math. Comp. 57 (1991), no. 195, 259–271.
R. Morelli, The K-theory of a toric variety, Adv. Math. 100 (1993), no. 2, 154–182.
M. Mustata, Vanishing theorems on toric varieties, Tohoku Math. J. (2) 54 (2002), no. 3, 451–470.
P.-E. Paradan, Jump formulas in Hamiltonian geometry, 2004, arXiv math 0411306, to appear in Geometric Aspects of Analysis and Mechanics, Proceedings of the conference in honor of H. Duistermaat, Utrecht 2007.
A. Szenes and M. Vergne, Residue formulae for vector partitions and Euler-Maclaurin sums, Formal power series and algebraic combinatorics — Scottsdale, AZ, 2001, Adv. in Appl. Math., vol. 30, 2003, arXiv math 0202253, pp. 295–342.
A. N. Varchenko, Combinatorics and topology of the arrangement of affine hyperplanes in the real space, Functional Anal. Appl. 21, no. 1 (1987), 9–19, Russian original publ. Funktsional. Anal. i Prilozhen. 21 (1987), no. 1, p.11–22.
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Berline, N., Vergne, M. (2012). Analytic continuation of a parametric polytope and wall-crossing. In: Bjorner, A., Cohen, F., De Concini, C., Procesi, C., Salvetti, M. (eds) Configuration Spaces. CRM Series. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-431-1_6
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DOI: https://doi.org/10.1007/978-88-7642-431-1_6
Publisher Name: Edizioni della Normale, Pisa
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