On a generalization of Bochner's tube theorem for generic CR-submanifolds (Q1100592)

This paper concerns a simple proof of the microlocal version of Bochner's tube theorem for generic CR-manifolds published by \textit{M. S. Baouendi} and \textit{F. Trèves} in Indiana Univ. Math. J. 31, 885--895 (1982; Zbl 0505.32013). From the text: Let \(N\) be a real analytic generic submanifold in a complex manifold \(X\). Let \({}^SX=(X-N)\sqcup S_NX\) be the real monoidal transform attaching the spherical normal bundle \(S_NX\) to the center \(N\). The boundary value sheaf on \(S_NX\) of holomorphic functions on \(X-N\) is defined as \(\mathcal A=(j_* (\mathcal O_X|X-N))|S_NX\), where \(j\:X-N\hookrightarrow{}^SX\) is the inclusion. The main result is stated as follows. Theorem: Let \(U\) be an open connected subset of \(S_NX\). Then (i) \(\Gamma(U,\mathcal A)=\Gamma(\text{ch}(U),\mathcal A)\); (ii) if \(\text{ch}(U)=\tau^{-1}(\tau(U))\), then \(\Gamma(U,\mathcal A)=\Gamma (\tau(U),\mathcal O_X|N)\), where \(\tau\:S_NX\to N\) is the projection, and \(\text{ch}(U)\) is the convex hull of \(U\) in \(S_NX\).