  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.
