Hodge-Riemann polynomials

2025/06/20 by Qing Lu, Weizhe Zheng, Lu, Qing +1
Mathematics · #05E14 #14C17 (Primary) 05A20 #14J60 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2506.16992

openalex publication_date 2025/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Positivity in the shadow of Hodge index theorem
• Chern numbers on positive vector bundles and combinatorics
• Schur positivity of difference of products of derived Schur polynomials
• Lorentzian polynomials on cones
• Hard Lefschetz properties, complete intersections and numerical dimensions
• Hodge-Riemann Relations for Schur Classes in the Linear and Kähler Cases
• Logarithmic concavity of Schur and related polynomials
• Hodge-Riemann bilinear relations for Schur classes of ample vector bundles
• Enumeration of bounded lecture hall tableaux
• Lorentzian polynomials
• Mixed Hodge-Riemann bilinear relations and m-positivity
• Log-Concave Polynomials I: Entropy and a Deterministic Approximation\n Algorithm for Counting Bases of Matroids
• On the Lefschetz and Hodge-Riemann theorems
• Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations
• The mixed Hodge-Riemann bilinear relations for compact Kahler manifolds
• Flags, Schubert polynomials, degeneracy loci, and determinantal formulas

Related

• On Hodge-Riemann Cohomology Classes
• Hodge-Riemann Relations for Schur Classes in the Linear and Kähler Cases
• Hodge-Riemann bilinear relations for Schur classes of ample vector bundles
• Lefschetz theorems, Hodge-Riemann relations and Ample vector bundles
• A Solution to the Hodge Conjecture