## Complex structures on a K3 surface as a hyperkähler manifold

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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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.