Microlocal vanishing cycles and ramified Cauchy problems in the Nilsson class (Q2709820)

Let \(X\) be a complex manifold and \(H\) a complex hypersurface of \(X\). Let \({\mathcal O}^{\text{ram}}_{H|X}\) be the sheaf of holomorphic functions on \(X-H\) ramified along \(H\).NEWLINENEWLINENEWLINEThe author first studies the microlocal structure of the microlocal version \(\mu\hom(F^H,{\mathcal O}_X)\) of \({\mathcal O}^{\text{ram}}_{H|X}\), which is introduced by \textit{A. D'Agnolo} and \textit{P. Schapira} [Duke Math. J. 64, No. 3, 451-472 (1991; Zbl 0760.35003)], where \(F^H\) is a sheaf attached to \(H\) and \({\mathcal O}_X\) is the sheaf of holomorphic functions on \(X\). The results obtained enable the author to study the microlocal nearby vanishing cycle and also to give a totally new proof of a result due to D'Agnolo and Schapira.NEWLINENEWLINENEWLINEFinally, the author solves the ramified Cauchy problem for \({\mathcal D}\)-modules with regular singularities for the initial data in the Nilsson class of Deligne, where \({\mathcal D}_X\) denotes the sheaf of rings of holomorphic differential operators of finite order on \(X\).