Kobayashi pseudometric on hyperkahler manifolds

Abstract

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 conjecture for any hyperkähler 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 hyperkähler manifolds are semi-ample. We prove that the Kobayashi pseudometric vanishes for any hyperkähler manifold with $b_2\ge 13$ if the SYZ conjecture holds for all its deformations. This proves the Kobayashi conjecture for all K3 surfaces and their Hilbert schemes.

v3: 19 pages, some proofs updated, a few corrections and references added

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• Bridgeland's stability and the positive cone of the moduli spaces of stable objects on an abelian surface
• Non-algebraic Hyperkaehler manifolds
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Cited by in corpus (5)

• Teichmuller spaces, ergodic theory and global Torelli theorem
• Ergodic complex structures on hyperkahler manifolds: an erratum
• Regenerations and applications
• Kobayashi non-hyperbolicity of Calabi-Yau manifolds via mirror symmetry
• Holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds with Lagrangian fibrations and special Kahler geometry