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A hyperkähler manifold is a Riemannian manifold of real dimension $4k$ and holonomy group contained in $Sp(k)$. It is known that every hyperkähler manifold has a $2$-sphere $S^{2}$ of complex structures with respect to which the metric is Kähler.

A K3 surface is a hyperkähler manifold of real dimension $4$. It is classic in algebraic geometry that its complex structure is parametrized by $\mathcal{D}_{K3}/\Gamma$ the period domain mod some arithmetic group (you may want to impose polarization). Note that here we don't think of the K3 surface as a Riemannian manifold.

My question is, are there any relation between the $2$-sphere $S^{2}$ above and the moduli space $\mathcal{D}_{K3}/\Gamma$? For example, can the moduli space be foliated by such $S^{2}$?

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up vote 8 down vote accepted

These $2$-spheres are called 'twistor lines'. They indeed cover the moduli space (in the non-polarized case) : more precisely, any two points of the moduli space may be linked by a chain of twistor lines.

A reference where this is nicely explained (and used !) is Huybrecht's Bourbaki talk about Verbitsky's Torelli theorem : More precisely, Definition 3.3 gives a lattice-theoretic definition of twistor lines, the link with your description of twistor lines is made in paragraph 4.4, and the result I mentionned above is Proposition 3.7.

In the polarized case, no twistor line is included in the moduli space, as a general member is not projective : see remark 8.1 of This article is particularly interesting in this respect. Indeed, Charles and Markman prove the standard conjectures for some projective hyperkähler varieties (a statement peculiar to projective varieties) using deformations along a twistor line (hence using non-projective varieties).

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Thanks for your response, Oliver. Twistor space interpretation makes sense (although I am not familiar with it). I will take a look at the references. Many thanks. – Michel Sep 2 '12 at 0:49

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