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Abstract. Let F (z) = Re(P (z)) + h.o.t be such that M = (F = 0) defines a germ of real analytic Levi-flat at 0 ∈ C n , n ≥ 2, where P (z) is a homogeneous polynomial of degree k with an isolated singularity at 0 ∈ C n and Milnor number µ. We prove that there exists a holomorphic change of coordinate φ such that φ(M ) = (Re(h) = 0), where h(z) is a polynomial of degree µ + 1 and j k 0 (h) = P .

Abstract. We consider a holomorphic foliation F of codimension k ≥ 1 on a homogeneous compact Kähler manifold X of dimension n > k. Assuming that the singular set Sing(F ) of F is contained in an absolutely k-convex domain U ⊂ X, we prove that the determinant of normal bundle det(N F ) of F cannot be an ample line bundle, provided [n/k] ≥ 2k + 3. Here [n/k] denotes the largest integer ≤ n/k.

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We study the singular set of a codimension one holomorphic foliation on P 3 . We find a local normal form for these foliations near a codimension two component of the singular set that is not of Kupka type. We also determine the number of non-Kupka points immersed in a codimension two component of the singular set of a codimension one foliation on P 3 .

We study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two. We give some hypotheses to guarantee the existence of dicritical singularities of these objects. As consequence, we give some applications to holomorphic foliations tangent to real-analytic Levi-flat hypersurfaces with singularities in P 2 .2010 Mathematics Subject Classification. Primary 32V40 -32S65.

In this paper we study normal forms of Levi-flat hypersurfaces with singularities. We prove a result analogous to the Burns-Gong theorem for the existence of rigid normal forms of Levi-flat hypersurfaces which are defined by the vanishing of the real part of A k , D k , E k singularities.