On the comparison theorem for elementary irregular $$\mathcal {D}$$-modules.(English)Zbl 0858.32013

Let $$U$$ be a quasi-projective variety over $$\mathbb{C}$$, $$f$$ be a regular function on $$U$$ and $${\mathcal M}$$ a regular holonomic $${\mathcal D}_U$$-module $$({\mathcal D}_U$$ is the sheaf of algebraic differential operators on $$U)$$. Denote by $${\mathcal M}_f$$ the $${\mathcal D}_U$$-module obtained from $${\mathcal M}$$ by twisting by $$e^f$$. The main result is a comparison theorem: $H^k \bigl(U,DR ({\mathcal M}_f) \bigr)=H^k_\varphi \bigl(U^{an}, DR^{an} ({\mathcal M}) \bigr) \quad \text{for all }k.$ Here $$DR({\mathcal M}_f)$$ is the (algebraic) de Rham complex of $${\mathcal M}_f$$, $$U^{an}$$ is the complex analytic manifold underlying $$U$$ (and $$DR^{an} ({\mathcal M})$$ the corresponding de Rham complex), $$H^k$$ is the $$k$$-th hypercohomology group and $$\varphi$$ is the family of closed sets of $$U^{an}$$ on which $$e^{-f}$$ is rapidly decreasing.
A local version of the result is proved. As a consequence it is shown that the irregularity complex of $${\mathcal M}_f$$ (here $$f$$ is supposed to be extended to $$F:X\to \mathbb{P}^1$$ where $$X$$ is a compactification of $$U$$ and $$\eta$$ is the inclusion $$U\hookrightarrow X)$$ has the same characteristic function as the nearby cycle complex of $$R\eta_* DR^{an} {\mathcal M}$$ with respect to $$1/F$$.

MSC:

 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14F40 de Rham cohomology and algebraic geometry
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