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Locally quasi-homogeneous free divisors are Koszul free. (English) Zbl 1031.32006
Proc. Steklov Inst. Math. 238, No. 3, 72-76 (2002) and Tr. Mat. Inst. Steklova 238, 81-85 (2002).
Let \(X\) be an \(n\)-dimensional complex analytic manifold, \(D \subset X\) a hypersurface. A divisor \(D\) in \(X\) is locally quasi-homogeneous if at each point \(q\in D\), there are local coordinates centered at \(q\) with respect to which \(D\) has locally a weighted (with positive weight) homogeneous defining equation. The divisor \(D\) is called Koszul free at \(x\) if the exists a basis \(\{ \delta_1, \dots, \delta_n\}\) of \(\text{Der} (\log D)_x\) such that the sequence of symbols \(\{\sigma (\delta_1), \dots, \sigma (\delta_n)\}\) is regular in \(\text{Gr}_F\cdot ({\mathcal D})= \text{Gr}_F \cdot({\mathcal D}_x)_x\). Here \(\text{Der}(\log D)\) is the \({\mathcal O}_x\)-module of the logarithmic vector fields with respect to \(D\) [K. Saito, J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 265-291 (1980; Zbl 0496.32007)], \(\text{Gr}_F \cdot({\mathcal D}_x)\) the graded ring associated to the filtration by the order, and \(\sigma(P)\) the principal symbol of a differential operator \(P\).
The main result of this paper is (Th. 3.2) that every locally quasihomogeneous free divisor is Koszul free.
In particular every free divisor that is locally quasi-homogeneous at the complement of a discrete set is Koszul free.
The proof uses a previous result of the authors [Trans. Am. Math. Soc. 348, 3037-3049 (1996; Zbl 0862.32021)].
The authors also give a number of examples of Koszul free divisors. Let us mention the example of a Koszul free divisor, irreducible which is not with normal crossing, or has nontrivial factors.
There is also a discussion concerning the relations between several kinds of free divisors.
For the entire collection see [Zbl 1012.00018].

32C38 Sheaves of differential operators and their modules, \(D\)-modules
14C20 Divisors, linear systems, invertible sheaves
14F40 de Rham cohomology and algebraic geometry