Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 889–913 | Cite as

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.

Keywords

Mirror symmetry Toric varieties Polytopes Orbifold cohomology Spectrum of regular tame functions 

Mathematics Subject Classification

32S40 14J33 34M35 14C40 

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Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Université Côte d’Azur, CNRS, LJAD, Parc ValroseNice Cedex 2France

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