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

Abstract

Let $X$ be a $d$ dimensional projective manifold, $E$ be an ample vector bundle on $X$ and $0\le {\lambda }_N\le {\lambda }_{N-1}\le \cdots \le {\lambda }_1\le \mathrm{rank}(E)$ 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.

v2. Two principal changes are (1) a generalisation of higher-rank Khovanskii-Tessier inequalities to Schur classes (Theorem 1.4) and (2) the inclusion of an application to cones of Nef cycles on self-products of a very general principally polarized abelian surface (Section 6) v3. Improved statement on derived Schur classes. Answers to some previously asked questions and examples provided