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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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