Cauchy-Schwarz-type inequalities on K\"{a}hler manifolds-II

TL;DR

This paper extends classical inequalities to higher-dimensional compact Kähler manifolds using advanced bilinear relations, offering new insights into geometric inequalities and proposing related proportionality problems.

Contribution

It generalizes Khovanskii-Teissier inequalities to higher dimensions on Kähler manifolds using mixed Hodge-Riemann relations.

Findings

Established new Cauchy-Schwarz-type inequalities for Kähler manifolds.

Connected inequalities to the mixed Hodge-Riemann bilinear relations.

Proposed a proportionality problem related to the main inequalities.

Abstract

We establish in this note some Cauchy-Schwarz-type inequalities on compact K\"{a}hler manifolds, which generalize the classical Khovanskii-Teissier inequalities to higher-dimensional cases. Our proof is to make full use of the mixed Hodge-Riemann bilinear relations due to Dinh and Nguy$\hat{\text{e}}$n. A proportionality problem related to our main result is also proposed.