Topological boundary values and regular \(\mathcal D\)-modules
type: Item
instance of: scholarly article
title: Topological boundary values and regular \(\mathcal D\)-modules
zbMATH Open document ID: 0951.32007
DOI: 10.1215/S0012-7094-98-09308-5
author: Yves Laurent
author: Teresa Monteiro Fernandes
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: 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 t…
Mathematics Subject Classification ID: 32C38
Mathematics Subject Classification ID: 35A27
Mathematics Subject Classification ID: 58J15
zbMATH DE Number: 1425118
zbMATH Keywords: microlocal analysis
zbMATH Keywords: hyperfunction
zbMATH Keywords: microfunction
zbMATH Keywords: \(\mathcal D\)-module
zbMATH Keywords: geometric boundary value
zbMATH Keywords: Fuchsian and hyperbolic operator
zbMATH Keywords: geometric vanishing cycles
zbMATH Keywords: vanishing cycles of \(\mathcal D\)-module
MaRDI profile type: Publication
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full work available at URL: s0012-7094-98-09308-5
OpenAlex ID: W2006542010
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description: scientific article; zbMATH DE number 1425118