ON THE EMBEDDING OF LEVI-FLAT HYPERSURFACES IN THE COMPLEX PROJECTIVE PLANE (AND AN APPENDIX WITH L´ASZL´O LEMPERT)

Let L be a hypothetical smooth Levi flat hypersurface in CP2 and r the signed distance to L by means of the Fubini-Study metric g. Denote Lru = cru the second order elliptic equation for the infinitesimal Levi-flat deformations of L, where cr = dbJbr + br ∧ Jbr, br = ιXrdγr, Xr = gradgr/ ∥gradgr∥2 g, γr is the restriction of dcr to L and db is the differentiation along the leafs of the Levi foliation. Then −cr ≥ H as leaf-wise (1, 1)-forms, where H is the holomorphic bisectional curvature of CP2. We give also an example of a Levi-flat manifold L of dimension 3 verifying that there exists a (1, 0)-form α on L such that ∂α is a K¨ahler form on every leaf of the Levi foliation, but L is not embeddable in CP2.