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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].

##### MSC:
 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
##### Keywords:
$${\mathcal D}$$-module; duality; free divisors
Macaulay2