The module \(DF^s\) for locally quasi-homogeneous free divisors.

*(English)*Zbl 1017.32023Let \(X\) 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.

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.

Reviewer: Gheorghe Gussi (Bucureşti)

##### 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 |