Non-Measure Hyperbolicity of Complex K3 Surfaces

TL;DR

This paper proves that all K3 surfaces and their Hilbert schemes, as well as Enriques surfaces, exhibit non-measure hyperbolicity, extending previous specific cases to all such surfaces.

Contribution

It establishes the non-measure hyperbolicity for all K3 surfaces, their Hilbert schemes, and Enriques surfaces, generalizing earlier partial results.

Findings

Non-measure hyperbolicity holds for all K3 surfaces.

Hilbert schemes of points on K3 surfaces are non-measure hyperbolic.

Enriques surfaces are also non-measure hyperbolic.

Abstract

We show that the non-measure hyperbolicity of K3 surfaces -- which M. Green and P. Griffiths verified for certain cases in 1980 -- holds for all K3 surfaces. As a byproduct, we prove the non-measure hyperbolicity of any Hilbert schemes of points on K3 surfaces. We also obtain a new proof of the non-measure hyperbolicity of any Enriques surface.