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

TL;DR

This paper proves that hyperkahler manifolds with higher Picard rank are not algebraically hyperbolic, especially when Picard rank is at least 3 or under certain conjectural conditions, and relates automorphism groups to hyperbolicity.

Contribution

It establishes non-algebraic hyperbolicity of hyperkahler manifolds with Picard rank ≥ 3 and links automorphism group properties to algebraic hyperbolicity.

Findings

Hyperkahler manifolds with Picard rank ≥ 3 are non-algebraically hyperbolic.

Conditional non-hyperbolicity for Picard rank 2 assuming the SYZ conjecture.

Infinite automorphism groups imply algebraic non-hyperbolicity.

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.