On Hodge-Riemann Cohomology Classes

TL;DR

This paper demonstrates that Schur classes of nef vector bundles can be approximated by classes satisfying Hodge-Riemann-like relations, leading to new log-concavity results and insights into Lorentzian properties of Schur polynomials.

Contribution

It introduces a novel approach to approximate nef vector bundle classes with Hodge-Riemann-like classes, enabling new applications in log-concavity and polynomial Lorentzianity.

Findings

Schur classes of nef vector bundles are limits of classes with Hodge-Riemann-like properties

New log-concavity statements for characteristic classes of nef vector bundles

Normalized Schur polynomials are Lorentzian

Abstract

We prove that Schur classes of nef vector bundles are limits of classes that have a property analogous to the Hodge-Riemann bilinear relations. We give a number of applications, including (1) new log-concavity statements about characteristic classes of nef vector bundles (2) log-concavity statements about Schur and related polynomials (3) another proof that normalized Schur polynomials are Lorentzian.