Vanishing cycles and \(D\)-modules
Let X be a complex manifold, \(f: X\to {\mathbb{C}}\) be a non-constant holomorphic function on X and \(X_ 0:=f^{-1}(0)\). Analogous versions of the Deligne's nearby cycle functor \(\psi_ f\), vanishing cycle functor \(\phi_ f\) aod of natural projections \[ can: \psi_ f\to \phi_ f,\quad var: \phi_ f\to \psi_ f \] are given in this paper. Using the functors \(\phi\) and \(\psi\) defined by Kashiwara on the category of regular holonomic \({\mathcal D}_ X\)-modules with the property that \(\phi{\mathcal M}\) (resp. \(\psi{\mathcal M})\) agree with \(\phi_ f{\mathcal F}\) (resp. \(\psi_ f{\mathcal F})\), \({\mathcal F}=R\hom_{{\mathcal D}_ X}({\mathcal M},{\mathcal O}_ x),{\mathcal M}\in mod({\mathcal D}_ X)_{hr}\), the author prove that the functor \[ F:{\mathcal M}\to ({\mathcal M}[[f^{-1}]],\phi{\mathcal M}\rightleftarrows ^{c}_{v}\psi{\mathcal M},\psi('p)) \] defines an equivalence of categories. Here \(\pi: {\mathcal M}\to {\mathcal M}[[f^{-1}]]\) denotes the canonical map and \(c({\mathcal M})\), v(\({\mathcal M})\) are the natural morphisms corresponding with can and var. By the Riemann-Hilbert correspondance this corresponds to Verdier's extension theorem of perverse sheaves and can be used for the proof. The author prefers to give a proof based only on the language of \({\mathcal D}\)-modules without the Riemann-Hilbert correspondance.
regular holonomic \({\mathcal D}_ X\)-modules
scientific article; zbMATH DE number 4129186
scientific article