Moduli of K3 families over $\mathbb{P}^1$ and complex-hyperk\"ahler metrics induced by deformed twistor cycles

Abstract

We answer a question posed independently by Fels-Huckleberry-Wolf and Looijenga concerning the geometric meaning of small deformations of twistor cycles in the K3 period domain. These are shown to induce complex-hyperk\"ahler metrics on members of the families via Penrose's Non-linear Graviton construction. On the way to proving this result, we construct a Hausdorff fine moduli space for families of marked K3 surfaces over smooth rational curves in the K3 period domain. Over an open subset containing all twistor cycles we construct a family of such families, which is a universal small deformation for every twistor family. Whenever possible, we extend the results to higher-dimensional irreducible holomorphic-symplectic manifolds.