Examples of Harmonic Foliated Currents and Singular Levi-flats on the Projective Plane

Abstract

We provide examples of foliations on the complex projective plane $\CP^2$ carrying positive foliated harmonic currents whose supports coincide with singular Levi-flats which, in turn, can be chosen real-analytic (but non-algebraic) or merely continuous with fractal transverse nature. Furthermore, non-trivial examples as above can already be found among foliations of degree~$2$ and~$3$. In addition, the space of positive foliated harmonic currents for these foliations is fully characterised and it contains a unique harmonic (non-closed) current supported on the Levi-flat in question. Finally, we also provide examples of foliations carrying diffuse positive foliated closed currents related to a theorem due to Brunella as well as a general criterion for the existence of singular real analytic Levi-flats for Riccati foliations.