Vanishing Cycles of Irregular D-Modules

Abstract

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.