A Riemannian manifold is *hyperkähler*, if there are three complex structures $I,J,K$, which are all compatible with the Riemannian metric (i.e., $(v,Iw)$ defines a symplectic form and similarly for $J$ and $K$). Furthermore, we also need the complex structures to satisfy the quaternionic relations $I^2=J^2=K^2=-1$ and $IJ+JI=0=IK+KI=JK+KJ$.


I was able to find only very few examples of closed (compact without boundary) hyperkähler manifolds, basically complex tori and K3-surfaces.
In higher dimensions than complex dimension 2, there are also generalized Kummer varieties of tori. Based on K3-surfaces, there are Hilbert schemes and resolutions of singularities in some moduli spaces.




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I am searching for an example of a slightly more general setting:
Let $X$ be a closed Riemannian manifold and $J\_1,..., J\_r$ be complex structures, all compatible with the Riemannian metric. Furthermore, assume that the $J\_l$ to satisfy the relations $J\_l^2=-1$ and $J\_lJ\_k+J\_kJ\_l=0$ for $k\neq l$.


If we take $X$ to be a vector space of dimension $2^{4a+b}c$ with $c$ odd, I know the maximal number for $r$ is $8a+2^b-1$, i.e. the maximal number is always odd. This carries over to quotients of vector spaces, i.e. for tori and quotients of tori. These examples are flat.


This yields the question:
Is there a closed, non-flat manifold admitting more than three such complex structures?


If $X$ admits more than three complex structures, it also admits three such structures. Therefore every such manifold has to be a hyperkähler manifold, i.e. one of the above mentioned examples or a product of them.
Are there different examples and does one of them admit more complex structures than the 3 for hyperkähler?



Existence of closed manifolds with more than 3 linearly independent complex structures?
A manifold admitting a triple of complex structures satisfying quaternionic
relations also admits a torsion-free connection (called "Obata connection") preserving the quaternionic structure. Such a connection is unique. For a hyperkaehler manifold, the Obata connection coinsides with the Levi-Civita. Therefore, the manifolds with this structure have Levi-Civita connection which preserves the complex structures $J\_i$. From Berger's classification of holonomy it follows that they are products of hyperkaehler manifolds, and
actually flat if $i>3$ (because $J\_4$ must exchange the tangent bundles to the factors of the product).


