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Global spectra, polytopes and stacky invariants

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Abstract

Given a convex polytope, we define its geometric spectrum, a stacky version of Batyrev’s stringy E-functions, and we prove a stacky version of a formula of Libgober and Wood about the E-polynomial of a smooth projective variety. As an application, we get a closed formula for the variance of the geometric spectrum and a Noether’s formula for two dimensional Fano polytopes (polytopes whose vertices are primitive lattice points; a Fano polytope is not necessarily smooth). We also show that this geometric spectrum is equal to the algebraic spectrum (the spectrum at infinity of a tame Laurent polynomial whose Newton polytope is the polytope alluded to). This gives an explanation and some positive answers to Hertling’s conjecture about the variance of the spectrum of tame regular functions.

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Authors and Affiliations

  1. Université Côte d’Azur, CNRS, LJAD, Parc Valrose, 06108, Nice Cedex 2, France

    Antoine Douai

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  1. Antoine Douai
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Correspondence to Antoine Douai.

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A. Douai was partially supported by the Grant ANR-13-IS01-0001-01 of the Agence nationale de la recherche.

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Douai, A. Global spectra, polytopes and stacky invariants. Math. Z. 288, 889–913 (2018). https://doi.org/10.1007/s00209-017-1918-8

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  • DOI: https://doi.org/10.1007/s00209-017-1918-8

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