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Algebraic nonhyperbolicity of hyperkähler manifolds with picard rank greater than one

par Kamenova, Ljudmila;Verbitskiy, Misha
Référence New York journal of mathematics, 23, page (489-495)
Publication Publié, 2017

Article révisé par les pairs

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Résumé :

A projective manifold is algebraically hyperbolic if the de-gree 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 hyperkähler manifolds are not 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 hyperkähler manifold is infinite then it is algebraically nonhyperbolic.