Hodge-Riemann polynomials

TL;DR

This paper introduces Hodge-Riemann polynomials, demonstrating their properties and applications in algebraic geometry, including confirming conjectures on Schur polynomials and their log-concavity.

Contribution

It defines Hodge-Riemann polynomials, proves their key properties, and applies them to confirm conjectures about Schur polynomial log-concavity.

Findings

Schur classes of ample vector bundles satisfy Hodge-Riemann relations under certain conditions

Hodge-Riemann polynomials produce cohomology classes with Hodge-Riemann properties

Derivative sequences of Schur polynomial products are Schur log-concave

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.