Do hyperKahler manifolds live in quaternionic-Kahler families? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T05:46:12Z http://mathoverflow.net/feeds/question/13188 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13188/do-hyperkahler-manifolds-live-in-quaternionic-kahler-families Do hyperKahler manifolds live in quaternionic-Kahler families? Marty 2010-01-27T21:11:27Z 2010-07-29T20:33:05Z <p>A geometry question that I thought about more seriously a few years ago... thought it'd be a good first question for MO.</p> <p>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.</p> <p>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.</p> <p>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:</p> <p>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.</p> <p>A <strong>family of hyperKahler manifolds</strong> over $X$ should be (I think) a fibration of complex manifolds $\pi: E \rightarrow Z$, such that:</p> <ol> <li>Each fibre $E_z = \pi^{-1}(z)$ is a hyperKahler manifold $(M_z, J_z)$ with distinguished integrable complex structure $J_z$. </li> <li>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).</li> </ol> <p>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 $Z$. </p> <p>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!)</p> http://mathoverflow.net/questions/13188/do-hyperkahler-manifolds-live-in-quaternionic-kahler-families/33846#33846 Answer by Simon Salamon for Do hyperKahler manifolds live in quaternionic-Kahler families? Simon Salamon 2010-07-29T20:33:05Z 2010-07-29T20:33:05Z <p>What you suggested makes sense. You propose to replace the $P^1$ fibre by the <em>twistor space</em> 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".</p> <p>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 <em>twistor space fibres</em> 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 $S^{2n-2}$!</p>