Log-concavity of polynomials arising from equivariant cohomology

TL;DR

This paper investigates the log-concavity of polynomials associated with equivariant cohomology classes of matrix subvarieties, establishing their covolume polynomial structure and connections to Lorentzian polynomials.

Contribution

It proves that these polynomials are covolume polynomials for a broad class of torus actions and links their dual generators to Lorentzian polynomials, extending classical results.

Findings

Polynomials are covolume polynomials in the studied setting.

Macaulay dual generators can be Lorentzian polynomials.

Provides a characteristic-free extension of cohomology ring descriptions.

Abstract

We study the equivariant cohomology classes of torus-equivariant subvarieties of the space of matrices. For a large class of torus actions, we prove that the polynomials representing these classes (up to suitably changing signs) are covolume polynomials in the sense of Aluffi. We study the cohomology rings of complex varieties in terms of Macaulay inverse systems over $\mathbb{Z}$. As applications, we show that under certain conditions, the Macaulay dual generator is a denormalized Lorentzian polynomial in the sense of Br\"and\'en and Huh, and we give a characteristic-free extension (over $\mathbb{Z}$) of the result of Khovanskii and Pukhlikov describing the cohomology ring of toric varieties in terms of volume polynomials.