Compact fibrations with hyperk\"ahler fibers

TL;DR

This paper proves that families of hyperk"ahler manifolds over simply connected bases have very limited variation, with essential dimension at most one, extending similar results to complex tori.

Contribution

It establishes a bound on the essential dimension of hyperk"ahler families over simply connected bases, a new result in the deformation theory of complex manifolds.

Findings

Essential dimension of hyperk"ahler families is at most 1.

Similar bound is shown for families of complex tori.

Results constrain the variation of hyperk"ahler manifolds in families.

Abstract

Essential dimension of a family of complex manifolds is the dimension of the image of its base in the Kuranishi space of the fiber. We prove that any family of hyperk\"ahler manifolds over a compact simply connected base has essential dimension not greater than $1$. A similar result about families of complex tori is also obtained.