On fibrations and measures of irrationality of hyper-K\"ahler manifolds

TL;DR

This paper investigates the structure of fibrations and rational maps on hyper-K"ahler manifolds, focusing on fiber genus and extending known results to broader classes of fourfolds.

Contribution

It provides new results on the fibers and images of rational maps from hyper-K"ahler manifolds and generalizes O'Grady's findings to more general fourfolds.

Findings

Results on the fibers and images of rational maps from hyper-K"ahler manifolds

Determination of minimal genus of fibers in fibrations into curves

Extension of O'Grady's results to broader hyper-K"ahler fourfolds

Abstract

We prove some results on the fibers and images of rational maps from a hyper-K\"ahler manifold. We study in particular the minimal genus of fibers of a fibration into curves. The last section of this paper is devoted to the study of the rational map defined by a linear system on a hyper-K\"ahler fourfold satisfying numerical conditions similar to those considered by O'Grady in his study of fourfolds numerically equivalent to $K3^{[2]}$. We extend his results to this more general context.