Kobayashi and Algebraic Non-Hyperbolicity of Hyperkaehler Manifolds

In this 52-minute lecture, Ljudmila Kamenova from Stony Brook University explores the Kobayashi and algebraic non-hyperbolicity properties of hyperkaehler manifolds. Discover the Kobayashi pseudometric on complex manifolds, defined as the maximal pseudometric where any holomorphic map from the Poincare disk to the manifold is distance-decreasing. Learn about Kobayashi's conjecture that this pseudometric vanishes on Calabi-Yau manifolds, implying they have "entire curves," and how this conjecture was proven for all K3 surfaces and many classes of hyperkaehler manifolds using ergodicity of complex structures. Examine the concept of algebraic hyperbolicity in projective manifolds and understand the proof that hyperkaehler manifolds are not algebraically hyperbolic under certain conditions, such as when the Picard rank is at least 3 or when the Picard rank is 2 and the SYZ conjecture holds. Also explore how an infinite automorphism group of a hyperkahler manifold implies algebraic non-hyperbolicity.