Curves of maximal moduli on K3 surfaces

TL;DR

The paper proves that every complex projective K3 surface contains infinitely many families of curves with maximal moduli variation, confirming conjectures and providing new algebraic geometric insights into their structure.

Contribution

It establishes the existence of infinitely many families of curves with maximal moduli variation on K3 surfaces, confirming a conjecture of Huybrechts and offering a new proof of Kobayashi's theorem.

Findings

Existence of infinitely many families of curves with maximal moduli on K3 surfaces

Every K3 surface contains a non-isotrivial genus 1 curve

Provides an algebraic proof that K3 surfaces lack global symmetric differential forms

Abstract

We prove that if $X$ is a complex projective K3 surface and $g>0$, then there exist infinitely many families of curves of geometric genus $g$ on $X$ with maximal, i.e., $g$-dimensional, variation in moduli. In particular every K3 surface contains a curve of geometric genus 1 which moves in a non-isotrivial family. This implies a conjecture of Huybrechts on constant cycle curves and gives an algebro-geometric proof of a theorem of Kobayashi that a K3 surface has no global symmetric differential forms.