Gaps in the support of canonical currents on projective K3 surfaces

TL;DR

This paper constructs examples of canonical positive currents on projective K3 surfaces that are not supported on the entire surface, revealing gaps in the support of such currents.

Contribution

It provides the first known examples of canonical currents on projective K3 surfaces with support gaps, using a novel Zassenhaus-type estimate for automorphism commutators.

Findings

Existence of canonical currents not fully supported on projective K3 surfaces.

Construction method based on automorphism commutator estimates.

Currents have vanishing self-intersection and are unique in their cohomology classes.

Abstract

We construct examples of canonical closed positive currents on projective K3 surfaces that are not fully supported on the complex points. The currents are the unique positive representatives in their cohomology classes and have vanishing self-intersection. The only previously known such examples were due to McMullen on non-projective K3 surfaces and were constructed using positive entropy automorphisms with a Siegel disk. Our construction is based on a Zassenhaus-type estimate for commutators of automorphisms.