* 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.



Is the wedge product of two harmonic forms harmonic?
It is easy to construct examples on Riemann surfaces of genus $>1$. Take any surface like this. Let $A$ and $B$ be two harmonic $1$-forms, that are not proportional. Then $A \wedge B$ is non-zero, but it vanishes at some point, since both $A$ and $B$ have zeros. At the same time a harmonic $2$-form on a Riemann surface is constant.
Explicit examples of $1$-forms on Riemann surfaces can be obtained as real parts of holomorphic $1$-forms.


Note of course that the above example is complex, and Einstein just take the standard metric of curvature $-1$. If you want an example on a Ricci flat manifold you should take a $K3$ surface. It is complex and admits a Ricci flat metric. Now, its second cohomology has dimension $22$. Now it should be possible to find two anti-self-dual two-forms whose wedge product vanishes at one point on $K3$ but is not identically zero. This is because the dimension of the space of self dual forms is 19 which is big enought to get vanishing at one point 


