Tuboids of \(\mathbb{C}^ n\) with cone property and domains of holomorphy (Q1184883)

Let \(X\) be a \(C^ \infty\)-manifold, \(M\) a closed submanifold and \(\Omega\) an open set of \(M\). We introduce a class of domains \(U\) of \(X\) called \(\Omega\)-tuboid. Then we take a complex of sheaves \({\mathcal F}\) on \(X\) and denote by \(\mu_ \Omega({\mathcal F})\) the microlocalization of \({\mathcal F}\) along \(\Omega\). We take a closed convex proper cone \(\lambda\) of \(T^*_ MX\) and describe the stalk of \(\mathbb{R}\pi_ *\mathbb{R}\Gamma_ \lambda\mu_ \Omega({\mathcal F})_{T^*_ MX}\) by means of cohomology groups of \({\mathcal F}\) over \(\Omega\)-tuboid \(U\) with profile \(\gamma=\text{int} \lambda^{oa}\). Let us take \(X=\mathbb{C}^ n\), \(M=\mathbb{R}^ n\) and \(\Omega\) an open convex set in \(M\). We can prove that there is a fundamental system of domains of holomorphy in the class of \(\Omega\)-tuboids with a prescribed profile. By this tool we prove a decomposition theorem for the microsupport at the boundary obtained by Schapira.