On microfunctions at the boundary along CR manifolds

Abstract

Let X be a complex analytic manifold, \(M \subset X\)a C2 submanifold, \(\Omega \subset M\) an openset with C2 boundary \(S = \partial \Omega \).Denote by \(\mu _M (\mathcal{O}_X )\) (resp. \(\mu _M (\mathcal{O}_X )\)) the microlocalization along M (resp. \(\Omega \)) of the sheaf \(\mathcal{O}_X \) of holomorphic functions.In the literature (cf. [A-G], [K-S 1,2])one encounters two classical results concerning the vanishing of the cohomology groups\(H^j \mu _M (\mathcal{O}_X )_p {\text{ for }}p \in \dot T_M^* X\).The most general gives the vanishing outside a range of indices j whose length is equal to\(s^0 (M,p)\) (with \(s^{ + , - ,0} (M,p)\) being the number of respectively positive, negative and null eigenvalues for the‘microlocal’ Levi form \(L_M (p)\)).The sharpest result gives the concentration in a single degree, provided that the difference\(s^ - (M,p\prime ) - \gamma (M,p\prime )\) is locally constant for \(p\prime \in T_M^* X\) near p (with\(\gamma (M,p) = {\text{ dim}}^C (T_M^* X \cap iT_M^* X)_z \) for z the base point of p).The first result was restated for the complex \(\mu _\Omega (\mathcal{O}_X )\)in [D'A-Z 2], in the case codim\(_M S = 1\)We extend it here to any codimension and moreover we also restate for \(\mu _\Omega (\mathcal{O}_X )\) the second vanishing theorem.We also point out that the principle of our proof, related to a criterion for constancy of sheaves due to [K-S 1], is a quite new one.