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Vanishing Cycles of Irregular D-Modules

Published online by Cambridge University Press:  04 December 2007

YVES LAURENT
Affiliation:
Université de Grenoble, Institut Fourier,Laboratoire de Mathématiques, UMR 5582 CNRS/UJF, BP 74, 38402 St. Martin d‘Hères Cedex, France
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Abstract

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Considering a holonomic ${\mathcal D}$-module and a hypersurface, we define a finite family of ${\mathcal D}$-modules on the hypersurface which we call modules of vanishing cycles. The first one had been previously defined and corresponds to formal solutions. The last one corresponds, via Riemann-Hilbert, to the geometric vanishing cycles of Grothendieck-Deligne. For regular holonomic ${\mathcal D}$-modules there is only one sheaf and for non regular modules the sheaves of vanishing cycles control the growth and the index of solutions. Our results extend to non holonomic modules under some hypothesis.

Type
Research Article
Copyright
© 1999 Kluwer Academic Publishers