2013/08/31 by Ljudmila Kamenova, Steven Lu, Misha Verbitsky · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Complex manifold #Conjecture #Curvature #Fibration #Geometry #Geometry and complex manifolds #Hermitian manifold #Holomorphic function #Hyperkähler manifold #Kähler manifold #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Pure mathematics #Scalar curvature #math.AG #math.DG
paper · pdf · doi:10.1112/jlms/jdu038
published as J. London Math. Soc. (2014) 90 (2): 436-450 · v3: 19 pages, some proofs updated, a few corrections and references added
openalex publication_date 2014/07/14 · arxiv created 2021/04/01 · arxiv updated 2021/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 b2≥ 13 if the SYZ conjecture holds for all its deformations. This proves the Kobayashi conjecture for all K3 surfaces and their Hilbert schemes.