Microlocalization of \({\mathcal O}_X\) along dihedral Lagrangians (Q1358726)

Let \(X\) be a complex manifold, \(T^*X\rightarrow X\) the cotangent bundle to \(X\), \(\sigma\) the canonical two-form on \(T^*X\), \(\Lambda_1, \Lambda_2\) two \(R\)-Lagrangian conic submanifolds of \(T^*X\). It is assumed that the intersection \(\Lambda_1\cap\Lambda_2\) is regular in a neighborhood of a point \(p\) and the intersection of the tangent planes at \(p\), \(\lambda_i(p)=T_p\Lambda_i\), \(\lambda_1(p)\cap\lambda_2(p)\) is of codimension one at \(\lambda_1(p)\). Then it is known that there exists a complex symplectic transformation \(\chi_1\) which interchanges \(\Lambda_1, \Lambda_2\) with the conormal bundles \(T_{M_1}^*X, T_{M_2}^*X\) to two hypersurfaces \(M_1,M_2\subset X\) whose Levi-forms are positive-semidefinite at \(q\in \lambda_1(p)\) [see \textit{A. D'Agnolo} and \textit{G. Zampieri}, Commun. Partial Differ. Equations 17, No. 5/6, 989-999 (1992; Zbl 0770.32014)]. In the paper under review, it is shown that there exists another symplectic transformation \(\chi_2\) such that the Levi-form of one hypersurface is positive-semidefinite, whereas the other has one negative eigenvalue. Let \(\Lambda=\Lambda_1^+\cup\Lambda_2^+\), where \(\lambda_1^+\) (\(\lambda_2^+\)) denotes one half-part of \(\Lambda_1\) (\(\Lambda_2\)) with boundary \(\Lambda=\Lambda_1^+\cup\Lambda_2^+\). Then in the work is proved that \(\Lambda\) can be reduced to the conormal bundle \(T_Y^*X\) to a \(C^1\)-hypersurface \(Y\) of \(X\) by one and only one of the transformations \(\chi_1\), \(\chi_2\). The proof is based on analysis of the shift of simple sheaves along the \(\Lambda_i\)'s under the action of quantizations of the \(\chi_i\)'s. The cohomology of the complex of microfunctions along \(\Lambda\) is studied in order to improve previous results of the author on the existence for \(\overline{\partial}\) on the dihedrons of \(\mathbb{C}^n\).