Article Dans Une Revue Advances in Mathematics Année : 2024

Examples of harmonic foliated currents and singular Levi-flats on the projective plane

Résumé

We provide examples of foliations on the complex projective plane 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 characterized 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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Dates et versions

hal-04930840 , version 1 (05-02-2025)

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Mohamad Alkateeb, Julio Rebelo. Examples of harmonic foliated currents and singular Levi-flats on the projective plane. Advances in Mathematics, 2024, 452, pp.109792. ⟨10.1016/j.aim.2024.109792⟩. ⟨hal-04930840⟩
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