In general, the product of two harmonic differential forms is not harmonic. However, for Kähler manifolds, the product of two harmonic forms is harmonic. What is a counterexample for the first statement and how do I prove the second? Thanks.
Products of harmonic forms are not always harmonic, not even on a Kähler manifold. The simplest counterexample is on a Riemann surface of genus at least two (which is obviously Kähler with respect to any metric). All harmonic one-forms have zeros because the Euler characteristic is non-zero, so the wedge product of two harmonic one-forms has zeros. But the only harmonic two-forms are the constant multiples of the Riemannian volume form, which have no zeros (unless it is the zero multiple).
This and related issues are discussed in the paper
D. Kotschick: On products of harmonic forms, Duke Math. Journal 107 (2001), 521--531.