By a recent result of Gauduchon, Moroianu and Semmelmann if a positive quaternion Kahler manifold admits any almost complex structure then it must be the complex Grassmannians $Gr_2(\mathbb C^{n+2})$. The latter is not hypercomlex so there are no hypercomplex manifolds which are also positive Quaternion Kahler. Note that it's even easier if you assume that the hypercomplex structure is compatible with the quaternion Kahler structure which the above argument does not assume but I think the original question did assume. That implies that the twistor bundle is trivial which is well-known not to be possible for positive quaternion Kahler manifolds. I don't know what happens in the negative quaternion Kahler case but I suspect there are no examples there either at least among compact ones. So I would guess that any compact hypercomplex manifold with holonomy in $Sp(n)\cdot Sp(1)$ must be hyper-Kahler.
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By a recent result of Gauduchon, Moroianu and Semmelmann if a positive quaternion Kahler manifold admits any almost complex structure then it must be the complex Grassmannians $Gr_2(\mathbb C^{n+2})$. The latter is not hypercomlex so there are no hypercomplex manifolds which are also positive Quaternion Kahler. I don't know what happens in the negative quaternion Kahler case but I suspect there are no examples there either at least among compact ones. So I would guess that any compact hypercomplex manifold with holonomy in $Sp(n)\cdot Sp(1)$ must be hyper-Kahler. |
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