  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.
