# Hodge-Riemann bilinear relations for Schur classes of ample vector bundles

## TL;DR

This paper proves Hodge-Riemann bilinear relations for Schur classes of ample vector bundles on projective manifolds, leading to new inequalities among characteristic classes and extending classical results.

## Contribution

It establishes the Hard Lefschetz property and Hodge-Riemann relations for Schur classes of ample vector bundles, a novel extension in algebraic geometry.

## Key findings

- Schur classes satisfy Hard Lefschetz property
- Schur classes obey Hodge-Riemann bilinear relations
- Derived new inequalities for characteristic classes

## Abstract

Let $X$ be a $d$ dimensional projective manifold, $E$ be an ample vector bundle on $X$ and $0\le \lambda_N\le \lambda_{N-1} \le \cdots \le \lambda_1 \le \operatorname{rank}(E)$ be a partition of $d-2$. We prove that the Schur class $s_{\lambda}(E)\in H^{d-2,d-2}(X)$ has the Hard Lefschetz property and satisfies the Hodge-Riemann bilinear relations. As a consequence we obtain various new inequalities between characteristic classes of ample vector bundles, including a higher-rank version of the Khovanskii-Teissier inequalities.