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Free divisors and duality for \(\mathcal D\)-modules. (English) Zbl 1039.32011
Proc. Steklov Inst. Math. 238, No. 3, 88-96 (2002) and Tr. Mat. Inst. Steklova 238, 97-105 (2002).
The relationship between \({\mathcal D}\)-modules and free divisors is considered in this paper. The authors prove a new duality formula between two \({\mathcal D}\)-modules associated to a class of free divisors on \(\mathbb{C}^n\) and give some applications. The main result of this paper is the following. If \(D\subset\mathbb{C}^n\) is a free locally quasi-homogeneous divisor, then the natural morphism \(\phi_D: \widetilde M^{\log D}\to{\mathcal O}[*D]\) is an isomorphism. In particular, \(M^{\log D}\) and \(\widetilde M^{\log D}\) are regular holonomic.
Here, the sheaf \({\mathcal O}[*D]\) is of meromorphic functions with poles along \(D\) and is naturally a left coherent \({\mathcal D}\)-module. The \({\mathcal D}\)-module \(\widetilde M^{\log D}\) is defined as follows. The ideal \(I^{\log D}\subset D\) is a left ideal generated by the logarithmic vector fields \(\text{Der}(\log D)\). It is a coherent sheaf of ideals in \(D\). The module \(M^{\log D}\) is defined by the quotient \({\mathcal D}/I^{\log D}\). We denote by \(\widetilde I^{\log D}\) the coherent sheaf of ideals of \({\mathcal D}\) generated by the set \(\{\delta+ a\mid\delta\in I^{\log}\) and \(\delta(f)= af\}\) for each local equation \(f\) of \(D\). Then we write \(\widetilde M^{\log D}={\mathcal D}/\widetilde I^{\log D}\). It has been proved to be holonomic.
For the entire collection see [Zbl 1012.00018].

32C38 Sheaves of differential operators and their modules, \(D\)-modules
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials