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Algebraic non-hyperbolicity of hyperkahler manifolds with Picard rank greater than one

Kamenova L., Verbitsky M.
A projective manifold is algebraically hyperbolic if the degree of any curve is 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¨ahler manifolds are non algebraically hyperbolic when the Picard rank is at least 3, or if the Picard rank is 2 and the SYZ conjecture on existence of Lagrangian fibrations is true. We also prove that if the automorphism group of a hyperk¨ahler manifold is infinite then it is algebraically non-hyperbolic.