Global spectra, polytopes and stacky invariants

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. * Partially supported by the grant ANR-13-IS01-0001-01 of the Agence nationale de la recherche. Mathematics Subject Classification 32S40, 14J33, 34M35, 14C40. Key words and phrases: Mirror symmetry, toric varieties, polytopes, orbifold cohomology, spectrum of regular tame functions. Y , with primitive generators v 1 , · · · , v r and associated divisors D 1 , · · · , D r . Put, for a subset J ⊂ I := {1, · · · , r}, D J := ∩ j∈J D j if J = ∅ and D J := Y if J = ∅ and define E st,P (z) := J⊂I

Stringy Chern classes of singular toric varieties and their applications

“…This generalization and its connection to the Libgober-Wood formula for smooth manifolds [LW90] (see also [Sal96]) has been considered independently in the recent preprint of Godinho, von Heymann, and Sabatini [GvHS16]. Independently some results of our paper were obtained in the preprint of Douai [Dou16] motivated by Hertling's conjecture about the variance of the spectrum of tame regular functions.…”

Ehrhart polynomials of polytopes and spectrum at infinity of Laurent polynomials

“…The main object of this paper is given by Definition 2.3, which is motivated by the description of the spectrum at infinity of a Laurent polynomial in Kouchnirenko's framework, see [5][6][7]13]. We first recall the construction.…”

“…Proposition 2.6 [5][6][7]14] Let P be a full-dimensional lattice polytope in N R containing the origin in its interior and let ν be its Newton function. Then,…”

“…Fortunately, if f is convenient and nondegenerate with respect to its Newton polytope in the sense of Kouchnirenko [10], and this is generically the case, the generating function Spec f (z) := μ i=1 z β i has a very concrete description; it is equal to the Hilbert-Poincaré series of the Jacobian ring of f graded by the Newton filtration [7]. As noticed in [5,6], it follows from Kouchnirenko's work that…”

“…The main difference is that the Newton polytope of a polynomial does not contain the origin as an interior point and that we have to deal with a non-complete situation. This can be applied for instance to the Mordell-Pommersheim tetrahedron, that is the convex hull of (0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c)) where a, b and c are positive integers, which is the Newton polytope of the Brieskorn-Pham polynomial In this context, other questions may arise: for instance, singularity theory predicts that the variance of the spectrum at infinity of a Laurent polynomial in (C * ) n is bounded below by n/12 (this is a global variant of C. Hertling's conjecture [9], see [6] and the references therein). By Proposition 5.1, this would give information about the distribution of the δ-vector of a reflexive polytope.…”