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Hodge-Riemann polynomials

Qing Lu, Weizhe Zheng

TL;DR

This work extends Hodge–Riemann theory to Schur classes of ample vector bundles on smooth projective varieties, under the vanishing hypothesis $H^{p-2,q-2}(X)=0$, and introduces Hodge–Riemann polynomials to systematize when partially symmetric polynomials yield HR-pairs. It connects these polynomials to Lorentzian and dually Lorentzian polynomials, showing that nonzero dually Lorentzian polynomials are precisely the ${}_{1,1}{ m HR}^k_{f 1^r}$ objects, and establishes broad preservation results under differentiation and multiplication by derived Schur polynomials. The results yield a robust toolkit for positivity: Schur and Schubert classes satisfy HR-type positivity on primitive cohomology, cone-class techniques extend to derived and twisted settings, and the derivative sequences of Schur polynomials exhibit Schur log-concavity, resolving conjectures of Ross, Wu, and others. Collectively, these contributions deepen the algebraic and geometric understanding of positivity in cohomology, with implications for combinatorics via HR polynomials and for geometric representation via degeneracy loci and derived constructions.

Abstract

We show that Schur classes of ample vector bundles on smooth projective varieties satisfy Hodge-Riemann relations on $H^{p,q}$ under the assumption that $H^{p-2,q-2}$ vanishes. More generally, we study Hodge-Riemann polynomials, which are partially symmetric polynomials that produce cohomology classes satisfying the Hodge-Riemann property when evaluated at Chern roots of ample vector bundles. In the case of line bundles and in bidegree $(1,1)$, these are precisely the nonzero dually Lorentzian polynomials. We prove various properties of Hodge-Riemann polynomials, confirming predictions and answering questions of Ross and Toma. As an application, we show that the derivative sequence of any product of Schur polynomials is Schur log-concave, confirming conjectures of Ross and Wu.

Hodge-Riemann polynomials

TL;DR

This work extends Hodge–Riemann theory to Schur classes of ample vector bundles on smooth projective varieties, under the vanishing hypothesis , and introduces Hodge–Riemann polynomials to systematize when partially symmetric polynomials yield HR-pairs. It connects these polynomials to Lorentzian and dually Lorentzian polynomials, showing that nonzero dually Lorentzian polynomials are precisely the objects, and establishes broad preservation results under differentiation and multiplication by derived Schur polynomials. The results yield a robust toolkit for positivity: Schur and Schubert classes satisfy HR-type positivity on primitive cohomology, cone-class techniques extend to derived and twisted settings, and the derivative sequences of Schur polynomials exhibit Schur log-concavity, resolving conjectures of Ross, Wu, and others. Collectively, these contributions deepen the algebraic and geometric understanding of positivity in cohomology, with implications for combinatorics via HR polynomials and for geometric representation via degeneracy loci and derived constructions.

Abstract

We show that Schur classes of ample vector bundles on smooth projective varieties satisfy Hodge-Riemann relations on under the assumption that vanishes. More generally, we study Hodge-Riemann polynomials, which are partially symmetric polynomials that produce cohomology classes satisfying the Hodge-Riemann property when evaluated at Chern roots of ample vector bundles. In the case of line bundles and in bidegree , these are precisely the nonzero dually Lorentzian polynomials. We prove various properties of Hodge-Riemann polynomials, confirming predictions and answering questions of Ross and Toma. As an application, we show that the derivative sequence of any product of Schur polynomials is Schur log-concave, confirming conjectures of Ross and Wu.

Paper Structure

This paper contains 21 sections, 104 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Degeneracy loci
  3. Schur classes
  4. Schubert classes
  5. Derived classes of $\mathbb{R}$-twisted vector bundles
  6. Positivity
  7. Chern classes
  8. Cone classes
  9. Hermitian forms and Hodge--Riemann pairs
  10. Definitions and first properties
  11. A differential criterion
  12. Schur classes and Schubert classes
  13. Top Chern classes
  14. Hodge--Riemann pairs
  15. Cone classes
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $E$ be an ample vector bundle of rank $e$ on a smooth projective variety $X$ of dimension $d$ over $\mathbb{C}$. Let $(p,q)$ be integers such that $p+q\le d$ and $H^{p-2,q-2}(X)=0$. Let $\lambda$ be a partition of $d-p-q$ satisfying $\lambda_1\le e$. Let $h=c_1(L)$, where $L$ is an ample line bu

Theorems & Definitions (229)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 219 more