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arXiv · 1604.02601

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

Ljudmila Kamenova·Misha Verbitsky
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Abstract

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 hyperkahler 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 hyperkahler manifold is infinite then it is algebraically non-hyperbolic.

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Ljudmila Kamenova, Misha Verbitsky. 2016-04-09. Algebraic non-hyperbolicity of hyperkahler manifolds with Picard rank greater than one. https://arxiv.org/abs/1604.02601

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