Cauchy–Schwarz-type inequalities on Kähler manifolds

In this note we investigate Cauchy–Schwarz-type inequalities for cohomology elements on compact Kähler manifolds, which can be viewed as generalizations of a classical case. We obtain, as a corollary, some Chern number inequalities when the Hodge numbers of Kähler manifolds satisfy certain restrictions. The same argument can also be applied to compact quaternion-Kähler manifolds with positive scalar curvature to obtain a similar result.