A geometry question that I thought about more seriously a few years ago... thought it'd be a good first question for MO.

I'm aware that there are a number of Torelli type theorems now proven for compact HyperKahler manifolds. Also, I think that Y. Andre has considered some families of HyperKahler (or holomorphic symplectic) manifolds in some paper.

But, when I see such a moduli problem studied, the data of a HyperKahler manifold seems to include a preferred complex structure. For example, a HyperKahler manifold is instead viewed as a holomorphic symplectic manifold. I'm aware of various equivalences, but there are certainly different amounts of data one could choose as part of a moduli problem.

I have never seen families of HyperKahler manifolds, in which the distinction between hyperKahler rotations and other variation is suitably distinguished. Here is what I have in mind, for a "quaternionic-Kahler family of HyperKahler manifolds:

Fix a quaternionic-Kahler base space $X$, with twistor bundle $Z \rightarrow X$. Thus the fibres $Z_x$ of $Z$ over $X$ are just Riemann spheres $P^1(C)$, and $Z$ has an integrable complex structure.

A **family of hyperKahler manifolds** over $X$ should be (I think) a fibration of complex manifolds $\pi: E \rightarrow Z$, such that:

- Each fibre $E_z = \pi^{-1}(z)$ is a hyperKahler manifold $(M_z, J_z)$ with distinguished integrable complex structure $J_z$.
- For each point $x \in X$, let $Z_x \cong P^1(C)$ be the twistor fibre. Then the family $E_x$ of hyperKahler manifolds with complex structure over $P^1(C)$ should be (isomorphic to) the family $(M, J_t)$ obtained by fixing a single hyperKahler manifold, and letting the complex structure vary in the $P^1(C)$ of possible complex structures. (I think this is called hyperKahler rotation).

In other words, the actual hyperKahler manifold should only depend on a point in the quaternionic Kahler base space $X$, but the complex structure should "rotate" in the twistor cover $Z$.

This sort of family seems very natural to me. Can any professional geometers make my definition precise, give a reference, or some reason why such families are a bad idea? I'd be happy to see such families, even for hyperKahler tori (which I was originally interested in!)