We construct a formal normal form for a class of real 2-codimensional submanifolds M ⊂ C N +1 defined near a CR singularity approximating the sphere. Our result gives a generalization of Huang-Yin's normal form in C 2 to a higher dimensional analogue case. Keywords: normal form, CR singularity, Fischer decompositionwhere ϕ ′ m,n z ′ , z ′ is a bihomogeneous polynomial of bidegree (m, n) in z ′ , z ′ satisfying the following normalization conditionsA few words about the construction of the normal form. We want to find a formal biholomorphic map sending M into a formal normal form. This leads us to study an infinite system of homogeneous equations by truncating the original equation. As in the paper [14] of Huang-Yin, this system is a semi-non linear system and is very hard to solve. We have then to use the powerful Huang-Yin's strategy and defining the weight of z k to be 1 and the weight of z k to be s − 1, for all k = 1 . . . , N . Since Aut 0 (M∞) is infinite-dimensional, it follows that the homogeneous linearized normalization equations (see sections 3 and 4) have nontrivial kernel spaces. By using the preceding system of weights and a similar argument as in the paper [14] of Huang-Yin , we are able to trace precisely how the lower order terms arise in non-linear fashion: The kernel space of degree 2t + 1 is restricted by imposing a normalization condition on ϕ ′ ts+1,0 (z) and the kernel space of degree 2t + 2 by imposing normalization conditions on ϕ ′ ts,0 (z). The non-uniqueness part of the lower degree solutions are uniquely determined in the higher order equations.

Abstract. Let (z 11 , . . . , z 1N , . . . , z m1 , . . . , z mN , w 11 , . . . , wmm ) be the coordinates in C mN +m 2. In this note we prove the analogue of the Theorem of Moser [24] in the case of the real-analytic submanifold M defined as followswhere W = w ij 1≤i,j≤m and Z = z ij 1≤i≤m, 1≤j≤N . We prove that M is biholomorphically equivalent to the model W = ZZ t if and only if is formally equivalent to it.

We construct a family of analytic discs attached to a real submanifold M ⊂ C N +1 of codimension 2 near a CR singularity. These discs are mutually disjoint and form a smooth hypersurface M with boundary M in a neighborhood of the CR singularity. As an application we prove that if p is a flat-elliptic CR singularity and if M is nowhere minimal at its CR points and does not contain a complex manifold of dimension (n − 2), then M is a smooth Levi-flat hypersurface. Moreover, if M is real analytic we obtain that M is real-analytic across the boundary manifold M .As an application of Theorem 1.1, we solve an open problem regarding the regularity of f given by Theorem 1.2 at q 1 , q 2 , proposed by Dolbeault-Tomassini-Zaitsev in [3]. By combining Theorem 1.1 and Theorem 1.2, we obtain the following result

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