Linear operators preserving volume polynomials

TL;DR

This paper characterizes linear operators that preserve volume polynomials, revealing a duality between homology and cohomology, and explores applications to matroid theory.

Contribution

It identifies covolume polynomials as the exact class of differential operators preserving volume polynomials, establishing a new duality and applications.

Findings

Covolume polynomials are the differential operators that preserve volume polynomials.

Establishes a duality between homology and cohomology in this context.

Provides applications to matroid theory.

Abstract

Volume polynomials measure the growth of Minkowski sums of convex bodies and of tensor powers of positive line bundles on projective varieties. We show that Aluffi's covolume polynomials are precisely the polynomial differential operators that preserve volume polynomials, reflecting a duality between homology and cohomology. We then present several applications to matroid theory.