\(q\)-pseudoconvex hypersurfaces through higher codimensional submanifolds of \({\mathbb C}^N\) (Q2781429)

Assume that \(S\) is a real analytic generic \(CR\) submanifold of \(\mathbb C^N\), of real codimension \(\ell>1\). The Levi form of \(S\) is a family of Hermitian quadratic forms on the analytic tangent space to \(S\), parametrized by the conormal vectors \(\xi\in T^*_S\mathbb C^N\). NEWLINENEWLINENEWLINEAssuming that the number \(p\) of positive and the number \(q\) of negative eigenvalues of the Levi form is constant for \(\xi\) in a neighborhood in \(T^*_S\mathbb C^N\) of a conormal direction \(\xi_0\) at a point \(x_0\) of \(S\), the author proves that there exists a real hypersurface through \(S\), having \(\xi_0\) as conormal direction at \(x_0\), whose Levi form near \(\xi_0\) has exactly \(p\) positive and \(q\) negative eigenvalues. NEWLINENEWLINENEWLINEHe uses this embedding result to prove results about the non-validity of the Poincaré lemma for the \(\bar\partial_S\) complex.