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!)
Best Answer
What you suggested makes sense. You propose to replace the $P^1$ fibre by the twistor space of an HK manifold M, so that the big total space would not only display separately the complex structures of M, but allow deformations of M to be parametrized by X. I think the real question is whether there exist sensible examples over a compact QK base like X$=S^4$ in which a consistent choice of complex structure on the varying HK manifolds is therefore not possible. I am not sure. The problem is that the construction looks a bit unwieldly, and experience dictates that it is more natural to look for bundles whose fibres are HK. In this sense, your idea is very close to a known (but in some sense simpler) construction that goes under the heading "Swann bundle" or "C map".
Let me add two comments in support of your question. First, the concept of a manifold foliated by HK manifolds (like $T^4$ or K3) is very powerful. This is most familiar in work on special holonomy, but here's a more classical construction: the curvature tensor at each point of a Riemannian 4-manifold can be used to construct a singular Kummer surface and an associated K3 (the intersection of 3 quadrics in $P^5$), but the complex structure is fixed so not twistorial. Second, escaping from quaternions, one sees twistor space fibres in the following situation: each fibre of the twistor space $SO(2n+1)/U(n)$ parametrizing a.c.s.'s on the sphere $S^{2n}$ can be identified with the twistor space of $S^{2n-2}$!