Extension of submanifolds of \(\mathbb{C}^n\) preserving the number of negative Levi eigenvalues (Q1382780)

Let \(S\) be a totally real submanifold of a complex manifold \(X\) which is \(C^2\). In a neighborhood of any point \(P\in S\), there exists a hypersurface \(M\) containing \(S\) which is the boundary of a strictly pseudoconvex domain. The author shows that if \(S\) is generic, then there exists a hypersurface \(M\) through \(S\) which has the same number of negative (or positive) Levi eigenvalues as \(S\) at a prescribed conormal (resp. at all common conormals when the rank of the Levi form \(L_S\) is assumed constant). The author applies this result to lift complex submanifolds from \(S\) to \(T_S^*X\), the conormal bundle of \(S\) in \(X\), when \(L_S\) is semidefinite of constant rank; this result was derived by \textit{E. Bedford} and \textit{J. E. Fornaess} [Duke Math. J. 48, 279-288 (1981; Zbl 0472.32007)] if the codimension is 1.