Algebraic nonhyperbolicity of hyperk?hler manifolds with Picard rank greater than one

摘要

A projective manifold is algebraically hyperbolic if the degree of any curveis bounded from above by its genus times a constant, which is independent from the curve.This is a property which follows from Kobayashi hyperbolicity.We prove that hyperk?hler manifolds are not algebraically hyperbolic when the Picard rank is at least 3, or ifthe Picard rank is 2 and the SYZ conjecture on existenceof Lagrangian fibrations is true.We also prove that if the automorphism group of a hyperk?hler manifold is infinite then it is algebraically nonhyperbolic.