Complex structures on a K3 surface as a hyperkähler manifold

Question

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 $S^{2}$?

Answer 1

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 : http://arxiv.org/abs/1106.5573. 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 http://arxiv.org/abs/1009.0413. 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).