On microfunctions at the boundary along CR manifolds

[摘要] Let $X$ be a complex analytic manifold, $M subset X$ a $C^2$ submanifold, $Omegasubset M$ an open set with $C^2$ boundary $S=partialOmega$. Denote by $mu_M({cal O}_X)$ (resp. $mu_Omega({cal O}_X)$) the microlocalization along $M$ (resp. $Omega$) of the sheaf ${cal O}_X$ 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$H^jmu_M({cal O}_X)_p$ for $pin dot{T}^*_MX$. 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 $p^{prime}in T^*_MX$ near $p$ (with $gamma(M,p)=dim^{m C}(T^*_MXcap iT^*_MX)_z$ for $z$ the base point of $p$).The first result was restated for the complex $mu_Omega({cal O}_X)$ in [D'A-Z 2], in the case ${m codim}_MS=1$. We extend it here to any codimension and moreover we also restate for $mu_Omega({cal O}_X)$ 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.

[学科分类] 数学（综合）