New York Journal of Mathematics
Volume 23 (2017) 489-495


Ljudmila Kamenova and Misha Verbitsky

Algebraic nonhyperbolicity of hyperkähler manifolds with Picard rank greater than one

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Published: April 4, 2017
Keywords: Hyperkähler manifold; algebraic hyperbolicity; SYZ conjecture
Subject: 53C26, 14J50

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 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.


Partially supported by RSCF grant 14-21-00053 within AG Laboratory NRU-HSE

Author information

Ljudmila Kamenova:
Department of Mathematics, 3-115, Stony Brook University, Stony Brook, NY 11794-3651, USA

Misha Verbitsky:
Laboratory of Algebraic Geometry, National Research University HSE, Faculty of Mathematics, Moscow, Russian Federation; also Université libre de Bruxelles, CP 218, Bd du Triomphe, 1050 Brussels, Belgium