TITLE: Is the wedge product of two harmonic forms harmonic?
QUESTION [18 upvotes]: Is the wedge product of two harmonic forms on a compact Riemannian manifold harmonic? I'm looking for a counter-example that the textbooks say exists. 
I would like to see a counter example that is on a complex manifold, Ricci-flat  (or Einstein) manifold or both, if it is at all possible. 
In general, I'm trying to understand the interaction between the wedge product, Hodge star and the Laplacian on forms and it's eigen-vectors, references will be much appreciated.

REPLY [8 votes]: Here is a cop-out answer (sorry in advance). Consider the 0-form $f(x)=x$ on $\mathbb{R}$. Clearly $f$ is harmonic, but $(f\wedge f)_{x}=x^2$ isn't. Since the wedge product is just the point-wise product on 0-forms, this is a valid example!