Kobayashi pseudometric on hyperk?hler manifolds

摘要

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré 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 for all hyperk?hler manifold with b_2 ≥ 7 that admits a deformation with a Lagrangian fibration and whose Picard rank is not maximal. The Strominger-Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperk?hler manifolds are semi-ample. We prove that the Kobayashi pseudometric vanishes for any hyperk?hler manifold with b_2 ≥ 7 if the SYZ conjecture holds for all its deformations. This proves the Kobayashi conjecture for all K3 surfaces and their Hilbert schemes.