Unobstructedness of hyperk\"ahler twistor spaces

TL;DR

This paper proves that certain families of hyperk"ahler manifolds, including twistor spaces, have unobstructed deformations when their period map is an embedding, ensuring smooth moduli spaces.

Contribution

It establishes unobstructedness of deformations for hyperk"ahler twistor spaces based on the properties of their period maps.

Findings

Families over the projective line have unobstructed deformations.

Families extend to universal families over open neighborhoods in the period domain.

Period map being an embedding implies unobstructedness.

Abstract

A family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.