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Volume polynomials

June Huh

TL;DR

Volume polynomials unify convex and algebraic geometry through their Lorentzian structure, encoding how volumes behave under Minkowski sums and how intersection numbers scale on projective varieties. The paper surveys realization problems across convex bodies, projective geometry, and polymatroids, foregrounding Branden–Huh's Lorentzian characterization and GHMSW's covolume framework to study linear operators preserving these polynomials. It connects Alexandrov–Fenchel-type inequalities to operator theory and showcases analytic volume polynomials via semipositive $(1,1)$-classes, while highlighting open questions about algebraicity, minors, and dual matroid behavior. The resulting toolkit offers a cohesive language for understanding which combinatorial and geometric data can be realized as volume polynomials, with implications for matroid theory, toric geometry, and algebraic geometry.

Abstract

Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss applications to the combinatorics of algebraic matroids. These notes are based on lectures given at the 2025 Summer Research Institute in Algebraic Geometry at Colorado State University.

Volume polynomials

TL;DR

Volume polynomials unify convex and algebraic geometry through their Lorentzian structure, encoding how volumes behave under Minkowski sums and how intersection numbers scale on projective varieties. The paper surveys realization problems across convex bodies, projective geometry, and polymatroids, foregrounding Branden–Huh's Lorentzian characterization and GHMSW's covolume framework to study linear operators preserving these polynomials. It connects Alexandrov–Fenchel-type inequalities to operator theory and showcases analytic volume polynomials via semipositive -classes, while highlighting open questions about algebraicity, minors, and dual matroid behavior. The resulting toolkit offers a cohesive language for understanding which combinatorial and geometric data can be realized as volume polynomials, with implications for matroid theory, toric geometry, and algebraic geometry.

Abstract

Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss applications to the combinatorics of algebraic matroids. These notes are based on lectures given at the 2025 Summer Research Institute in Algebraic Geometry at Colorado State University.
Paper Structure (5 sections, 12 theorems, 80 equations)

This paper contains 5 sections, 12 theorems, 80 equations.

Table of Contents

  1. Realization problems for projection volumes and homology classes
  2. Volume polynomials in convex geometry
  3. Volume polynomials in projective geometry
  4. Linear operators preserving volume polynomials
  5. Realization problems for polymatroids

Key Result

Theorem 1.1

The following conditions are equivalent for any vector of nonnegative real numbers $(p_{12},p_{13},p_{14},p_{23},p_{24},p_{34})$.

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Conjecture 1.6
  • Conjecture 1.7
  • Example 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 38 more