Hyperbolicity properties of moduli spaces of marked hyperk{\"a}hler manifolds

TL;DR

This paper investigates the hyperbolicity of moduli spaces of marked hyperkähler manifolds, showing vanishing Kobayashi pseudo-distance in certain directions and establishing restrictions on families over complex curves using Nevanlinna theory.

Contribution

It generalizes existing results to hyperkähler manifolds of arbitrary dimensions and noncompact curves, and links positivity of Hodge bundles to hyperbolicity properties.

Findings

Kobayashi pseudo-distance vanishes along certain directions

Existence of families over arbitrary curves with prescribed period maps

Restrictions on families over C with positive Hodge bundle

Abstract

We study the hyperbolicity properties of moduli spaces of marked hyperk{\"a}hler manifolds along directions corresponding to families having positivity properties for their Hodge bundle. In particular, we show that the Kobayashi pseudo-distance computed using disks tangent to these directions vanishes. As an intermediate step, we establish the existence of families of marked hyperk{\"a}hler manifolds over arbitrary curves having a prescribed period map to the corresponding period domain. This generalizes a recent theorem of Greb and Schwald [GS24] to the case of hyperk{\"a}hler manifolds of arbitrary dimensions and nonnecessarily compact curves. Finally, using Nevanlinna theory, we establish restrictions on families of hyperk{\"a}hler manifolds over C having positive Hodge bundle.