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The module $$DF^s$$ for locally quasi-homogeneous free divisors. (English) Zbl 1017.32023
Let $$X$$ be an $$n$$-dimensional complex analytic manifold, $$D \subset X$$ a divisor, $$\text{Der} (\log D)$$ the $${\mathcal O}_X$$-module of the logarithmic vector fields with respect to $$D$$, as introduced by K. Saito [J. Fac. Sci., Univ. Tokyo, Sect. IA 27, 265-291 (1980; Zbl 0496.32007)], and if $$f$$ is a local equation for $$D$$ or $$p\in X$$ $$\text{Der}(\log f)$$ the stalk at $$p$$ of $$\text{Der} (\log D)$$, whose elements are germs at $$p$$ of vector fields $$\delta$$, with $$\delta (f)\in m(f)$$.
A germ of divisor $$(D,p)\subset (X,p)$$ is quasi-homogeneous, in short q-h, if there are local coordinates: $$x_1,\dots, x_p\in {\mathcal O}_{X,p}$$ with respect to which $$(D,p)$$ has a weighted homogeneous defining equation (with strictly positive weights), and a divisor $$D$$ is called locally quasi-homogeneous if the germ $$(D,p)$$ is q-h for each $$p\in D$$. The divisor $$D$$ is free at $$p\in X$$ if $$\text{Der}(\log D)_p$$ is a free $${\mathcal O}$$-module (of rank $$n)$$, and it is Koszul-free at $$p$$ if it is free at $$p$$ and there exists a basis $$\{\delta_1, \dots, \delta_n)$$ of $$\text{Der}(\log D)$$ such that the sequence of symbols $$\{\sigma (\delta_1, \dots, \sigma (\delta_n)\}$$ is regular in $$\text{Gr}_F \bullet (D)_{X^p}$$, where the filtration $$F^\bullet$$ is the order filtration.
The authors find an explicit free resolution of $${\mathcal D}f^s$$, respectively $${\mathcal D}[(s)] f^s/ {\mathcal D}[s] f^{s+1}$$, where $$f$$ is a reduced equation of a locally q-h free divisor.
Let $$\Theta_f= \{\delta\in \text{Der}_\mathbb{C} ({\mathcal O})\mid \delta (f)=0\}$$ and denote by $$\widetilde\Theta$$ the $${\mathcal O}_X$$-submodule (and Lie algebra) of $$\text{Der} (\log D)$$ whose sections are vector fields annihilating $$f$$. $$I_f$$ will be the Jacobian ideal associated to $$f$$, finally denote by $$\Xi_f= \Theta_f \oplus( {\mathcal O}_f)$$.
The main result of the paper is: suppose that the equation $$f$$ and its Euler field $$E$$ are globally defined on $$X$$, $$E(q)\neq 0$$ for every $$q\in X\setminus \{p\}$$, $$\text{Der} (\log D)$$ is $${\mathcal O}_X$$-free (of rank $$n=\dim X)$$. Then
(1) The Koszul complex associated with $$\Theta_f \subset\text{Gr}_F\bullet ({\mathcal D})$$ is exact.
(2) The Spencer complex associated with $$\Xi_f\subset {\mathcal D}$$ is a free resolution of $${\mathcal D}[s] f^s/{\mathcal D}[s] f^{s+1}$$.
Moreover
(3) If $$f$$ is a reduced local equation of a locally q-h free divisor in $$\mathbb{C}^n$$ and $$\{\delta_1, \dots, \delta_{n-1}\}$$ a basic of module of vector fields vanishing on $$f$$, then the $$\delta_i$$ generate the ideal $$\text{Ann}_{\mathcal D}f^s$$.
The proof depends on the following interesting result: Every locally $$q-h$$ free divisor is Koszul free, whose proof is given in section 4 of this paper [this result was previously proved by the authors in Proc. Steklov Inst. Math. 238 (2002)].
Under the same hypotheses, the Spencer-complex for $$\Theta_f$$ is a resolution of $${\mathcal D}f^s$$ (the Spencer complex for $$\Theta_f$$ is the stalk of $$p$$ of the Spencer complex).
The last section of the paper gives some interesting examples and raises some problems.

##### MSC:
 32S20 Global theory of complex singularities; cohomological properties 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 32S45 Modifications; resolution of singularities (complex-analytic aspects) 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14F40 de Rham cohomology and algebraic geometry
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