Kobayashi pseudometric on hyperkahler manifolds

TL;DR

This paper proves Kobayashi's conjecture that the pseudometric vanishes on certain hyperk"ahler manifolds, including all K3 surfaces and their Hilbert schemes, by leveraging ergodic theory and the SYZ conjecture.

Contribution

It establishes the vanishing of the Kobayashi pseudometric on hyperk"ahler manifolds under specific conditions, connecting complex geometry, ergodicity, and the SYZ conjecture.

Findings

Kobayashi pseudometric vanishes on hyperk"ahler manifolds with certain fibrations.

Proves Kobayashi conjecture for all K3 surfaces and their Hilbert schemes.

Links the vanishing to the validity of the SYZ conjecture for deformations.

Abstract

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincar\'e disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture for any hyperk\"ahler manifold that admits a deformation with two Lagrangian fibrations and whose Picard rank is not maximal. The Strominger-Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperk\"ahler manifolds are semi-ample. We prove that the Kobayashi pseudometric vanishes for any hyperk\"ahler manifold with $b_2\geq 13$ if the SYZ conjecture holds for all its deformations. This proves the Kobayashi conjecture for all K3 surfaces and their Hilbert schemes.