Levi’s forms of higher codimensional submanifolds. (English) Zbl 0741.58045

It is proved that the number \(s(S,p)\), \(p\in\dot T^*_ SX\), defined as the Maslov index for three Lagrangian planes [see: M. Kashiwara and 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 M. Kashiwara and 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.
Reviewer: S.Dimiev (Sofia)


58J32 Boundary value problems on manifolds
58J15 Relations of PDEs on manifolds with hyperfunctions
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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