On Some Birational Invariants of Hyper-Kähler Manifolds

Abstract

Abstract We study in this article three birational invariants of projective hyper-Kähler manifolds: the degree of irrationality, the fibering gonality, and the fibering genus. We first improve the lower bound in a recent result of Voisin saying that the fibering genus of a Mumford–Tate very general projective hyper-Kähler manifold is bounded from below by a constant depending on its dimension and the second Betti number. We also study the relations between these birational invariants for projective $K3$ surfaces of Picard number $1$ and study the asymptotic behaviors of their degree of irrationality and fibering genus.