# zbMATH — the first resource for mathematics

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].

##### MSC:
 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14C20 Divisors, linear systems, invertible sheaves 14F40 de Rham cohomology and algebraic geometry