Levi's forms of higher codimensional submanifolds (Q1175692)

It is proved that the number \(s(S,p)\), \(p\in\dot T^*_ SX\), defined as the Maslov index for three Lagrangian planes [see: \textit{M. Kashiwara} and \textit{P. Schapira}, `Microlocal study of sheaves' (1985; Zbl 0589.32019)] coincides with the signature of the restriction on the complexified tangent space \(\mathbb{C} T_{z_ 0}S\), \(z_ 0\in S\), of the Levi form \(L_ M\), where \(M\) is a hypersurface of \(\mathbb{C}^ n\) and \(S\) is a submanifold of \(M\), i.e. \(s(S,p)=\hbox{sgn}(LM\mid \mathbb{C} T_{z_ 0}S)\). Obtaining a formula for the rank\((L_ M\mid \mathbb{C} T_{z_ 0}S)\), which includes the codimension of \(S\) in \(\mathbb{C}^ n\), the authors prove that, introduced by \textit{M. Kashiwara} and \textit{P. Schapira} [Invent. Math. 82, 579-592 (1985; Zbl 0626.58028)] the numbers \(s^ \pm(S,p)\) are respectively the positive and negative eigenvalues for \(L_ M\mid CT_{z_ 0}S\), which is of course true for \(S=M\). An application for microfunctions at the boundary are given.