Global spectra, polytopes and stacky invariants

TL;DR

This paper introduces a new geometric spectrum for convex polytopes, extending Batyrev's stringy E-functions, and establishes formulas connecting polytope invariants with algebraic and topological properties, including a proof of Hertling's conjecture.

Contribution

It defines a stacky geometric spectrum for polytopes, proves a formula relating it to E-polynomials, and applies these results to Fano polytopes and Hertling's conjecture.

Findings

Derived a closed formula for the variance of the geometric spectrum.

Established a Noether's formula for two-dimensional Fano polytopes.

Proved the equality of geometric and algebraic spectra for polytopes.

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). We also show that this geometric spectrum is equal to the algebraic spectrum of the polytope (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.