On length and product of harmonic forms in Kaehler geometry

Abstract

Motivated by understanding the limiting case of a certain systolic inequality we study compact Riemannian manifolds having all harmonic 1-forms of constant length. We give complete characterizations as far as K\"ahler and hyperbolic geometries are concerned. In the second part of the paper, we give algebraic and topological obstructions to the existence of a geometrically 2-formal K\"ahler metric, at the level of the second cohomology group. A strong interaction with almost K\"ahler geometry is to be noted. In complex dimension 3, we list all the possible values of the second Betti number of a geometrically 2-formal K\"ahler metric.