Examples of Harmonic Foliated Currents and Singular Levi-flats on the Projective Plane (Mohamad Alkateeb - Université Paul Sabatier)

Séminaire « Analyse complexe et équations différentielles »
Salle Kampé de Fériet

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.

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