On some birational invariants of hyper-K\"ahler manifolds

TL;DR

This paper investigates three birational invariants of projective hyper-K"ahler manifolds, improving bounds on the fibering genus and exploring their relationships and asymptotic behaviors, especially for K3 surfaces.

Contribution

It refines lower bounds on the fibering genus for hyper-K"ahler manifolds and analyzes the invariants' relationships and asymptotics for K3 surfaces.

Findings

Lower bound on fibering genus depends on dimension and Betti number.

Relations between invariants are studied for K3 surfaces.

Asymptotic behaviors of degree of irrationality and fibering genus are analyzed.

Abstract

We study in this article three birational invariants of projective hyper-K\"ahler 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-K\"ahler 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.