Vanishing cycles of \(D\)-modules. (English) Zbl 0799.32031

Let \(X\) be a complex manifold and \(i:Y \to X\) the inclusion of a smooth hypersurface. If \({\mathcal M}\) is a \({\mathcal D}_ X\) or \({\mathcal E}_ X\)- module, let \(\widetilde \Phi ({\mathcal M}) = i^* (\widetilde {\mathcal D}^ 2 \otimes {\mathcal M})\), where \(\widetilde {\mathcal D}^ 2\) is a sheaf of 2-microdifferential operators and \(i^*\) is the extension to these operators of the inverse image of \({\mathcal D}\)-modules. \(\widetilde\Phi({\mathcal M})\) is a sheaf on the conormal bundle \(\pi:T_ Y^*X\to Y\) and has a \(\pi^{-1} {\mathcal D}_ Y\)-module structure. In fact if \({\mathcal M}\) is specializable (i.e., a \(b\)-function exists for \({\mathcal M})\), then \(\widetilde \Phi ({\mathcal M})\) is coherent as a \(\pi^{- 1} {\mathcal D}_ Y\)-module on \(T_ Y^* X-Y\) and hence locally constant over \(Y\). The equation of \(Y\) gives a nonvanishing section \(\sigma:Y \to T_ Y^* X\) and a coherent \({\mathcal D}_ Y\)-module \(\Phi ({\mathcal M}) = \sigma^{-1} \widetilde \Phi ({\mathcal M})\). This module is proved to be the same as the module of vanishing cycles defined by Kashiwara, Malgrange and Sabbah. This new definition allows one to calculate the characteristic cycle of \(\widetilde \Phi ({\mathcal M})\) from the microcharacteristic cycle of \({\mathcal M}\).
In the case when \({\mathcal M}\) is specializable, the author also proves that in the case of an irregular singularity the sheaf \(\widetilde \Phi ({\mathcal M})\) does not change if \(\widetilde {\mathcal D}^ 2\) is replaced by a sheaf \(\widetilde {\mathcal D}^ 2(r)\) of 2-microdifferential operators with specified growth. Then, a Cauchy theorem relates the solutions of the system \({\mathcal M}\) in the sheaf of microfunctions to the holomorphic solutions of \(\widetilde \Phi ({\mathcal M})\). Combining these allows the author to evaluate the growth of solutions of \({\mathcal M}\). The results are extended further to nonsmooth hypersurfaces \(Y\).
The proofs make extensive use of earlier results of the author on 2-microlocalization and results on \({\mathcal D}\)-modules and are hard to understand without these papers at hand.
Reviewer: J.S.Joel (Kelly)


32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32C38 Sheaves of differential operators and their modules, \(D\)-modules
58J15 Relations of PDEs on manifolds with hyperfunctions
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