In 'panoramic view of Riemmannian geometry' when introducing hyperkähler manifolds, Berger states, informally, that a hyperkähler manifold is a Riemmannian manifold which is Kähler for more than one different almost complex structures.

I was wondering whereas this was a theorem or just a 'catchphrase'. In other words, my question is : if a Riemannian manifold is Kähler for two different (linearly independent) almost complex structures, is it hyperkähler ?

Of course on has to ask the metric to be irreducible, since things such as $\mathbb{S}^2\times M$ where $M$ is hyperkähler has at least 6 (!) independent almost complex structures. And being $4n+2$ dimensional, they can't be hyperkähler !

I think this is true in (real) dimension 4 for the following reason : the holonomy group $G$ of a Kähler $(M^4,g)$ is contained in $U(2)$. It leaves the Kähler form invariant. If there are two almost complex structures, there are two Kähler forms, so $G$ must act trivially on a 2-dimensional subspace of $\Lambda^2T_p^*M$. Since the holonomy representation of $U(2)$ only preserves the Kähler form, $G$ is a strict subgroup of $U(2)$ and is therefore contained in $SU(2)=Sp(1)$, hence $M$ is hyperkähler.

My knowledge of representation theory is too scarce to see if this carry on in higher dimensions.