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arXiv · 1308.5667

Kobayashi pseudometric on hyperkahler manifolds

Ljudmila Kamenova·Steven Lu·Misha Verbitsky
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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.

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Ljudmila Kamenova, Steven Lu, Misha Verbitsky. 2013-08-26. Kobayashi pseudometric on hyperkahler manifolds. https://doi.org/10.1112/jlms%2Fjdu038

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