Topological boundary values and regular \(\mathcal D\)-modules
The paper is devoted to the microlocal analysis of boundary value problems for hyperfunctions and microfunction solutions of complex differential systems regarded through the ``\(\mathcal D\)-module'' point of view, that is, defining the morphism of boundary values as functor between the hyperfunction solutions of a \(\mathcal D\)-module and the hyperfunction solutions of its vanishing and nearby cycles. The first aim is to prove a result analogous to \textit{H. Tahara} [Jap. J. Math., New Ser. 5, 245-347 (1979; Zbl 0431.35004)], in the framework of a \(\mathcal D\)-module of the paper by \textit{T. Monteiro Fernandes} [Compos. Math. 81, 121-142 (1992; Zbl 0754.58034)]. Working with a \(\mathcal D\)-module, the authors cannot assume that an operator is simultaneously Fuchsian and hyperbolic. Therefore, they separate the hypotheses and prove that if the modules is Fuchsian (more precisely, regular along a hypersurface), then there is a boundary value morphism, and if one adds a hypothesis of hyperbolicity, then it is an isomorphism. In fact, it appears that the condition of regularity itself is not necessary. If a geometrical condition on the characteristic variety of the \(\mathcal D\)-module is assumed, one can define a boundary value morphism, which is an isomorphism with the condition of hyperbolicity (which itself is a geometrical condition on the characteristic variety). In this context, the morphism has values in the geometric vanishing and nearby cycles of the sheaf of solutions of a given system. Now the regularity condition is used to identify these geometric vanishing cycles of solutions with solutions of a \(\mathcal D\)-module, called the vanishing cycles of the \(\mathcal D\)-module. Once these conditions are reduced to conditions on the characteristic variety of the \(\mathcal D\)-module, it appears that we may forget completely \(\mathcal D\)-modules and use the theory of sheaves of Kashiwara-Schapira. More precisely, the authors prove a theorem for the complex of sheaves of \(\mathbb C\)-vector spaces, the conditions on the characteristic variety becoming conditions on the microsupport of the complex. This boundary value morphism may be called a ``topological boundary value morphism'' as that of Schapira. Then the result on the \(\mathcal D\)-module is the special case where the complex of sheaves is the complex of holomorphic solutions of the \(\mathcal D\)-module. An easy application of the obtained results to the so-called ``ideally analytic'' solutions is given.
vanishing cycles of \(\mathcal D\)-module
scientific article; zbMATH DE number 1425118
scientific article