Testing the Logarithmic Comparison Theorem for Free Divisors

Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=uexm20 Experimental Mathematics ISSN: 1058-6458 (Print) 1944-950X (Online) Journal homepage: https://www.tandfonline.com/loi/uexm20 Testing the Logarithmic Comparison Theorem for Free Divisors F. J. Castro-Jiménez & J. M. Ucha-Enríquez To cite this article: F. J. Castro-Jiménez & J. M. Ucha-Enríquez (2004) Testing the Logarithmic Comparison Theorem for Free Divisors, Experimental Mathematics, 13:4, 441-449, DOI: 10.1080/10586458.2004.10504553 To link to this article: https://doi.org/10.1080/10586458.2004.10504553 Published online: 03 Apr 2012. Submit your article to this journal Article views: 16 View related articles Citing articles: 6 View citing articles Testing the Logarithmic Comparison Theorem for Free Divisors F. j. Castro-jimenez and j. M. Ucha-Enrfquez CONTENTS 1. Introduction 2. Spencer Divisors 3. How to Deduce that LCT Holds 4. How to Deduce that LCT Does Not Hold 5. On the Regularity of Logarithmic 1>.Modules 6. Conclusions Acknowledgments References 2000 AMSSubject Classification: Primary 14F50; Secondary 32C38, 32C35, 13Pxx, 68W30 Keywords: de Rham cohomology, Logarithmic Comparison Theorem, free divisors, Grobner bases We propose in this work a computational criterion to test if a free divisor DC C" verifies the Logarithmic Comparison Theorem (LCT); that is, whether the complex of logarithmic differential forms computes the cohomology of the complement of D inC n• ForSpencerfree divisors D == (I =0), we solve a conjecture about the generators of the annihilating ideal of 1/1 and make a conjecture on the nature of Euler homogeneous free divisorswhich verify LCT. In addition, we provideexamples of free divisors defined by weighted homogeneous polynomials that are not locally quasi-homogeneous. 1. INTRODUCTION Let Dbe a divisor (i.e., a hypersurface) in X := en. K. Saito introduced in [Saito 80] the complex n-(log D) of holomorphic differential forms with logarithmic poles along D. It is a sub complex of the meromorphic de Rham complex n-(*D) of meromorphic differential forms with poles along D. Let us denote by io the inclusion morphism n-(logD) ~ n-(*D). Grothendieck's Comparison Theorem [Grothendieck 66] proves that the last complex calculates the cohomology of the complement of D in X. It was proved in [Castro et al. 96] that if Dis alocally quasi-homogeneous free divisor then the Logarithmic Comparison Theorem (LCT) holds for D. We claim that LeT holds for Dif the morphism iDis a quasi-isomorphism, i.e., if io induces an isomorphism on cohomology. Let us denote by 0=0x the sheaf of holomorphic functions on X and take xE X. Denote by Der(Ox) the Ox-module of C-derivations of Ox (the elements in Der(Ox) are called vector fields). This yields the sheaf Der( 0) of vector fields on X. Following K. Saito [Saito 80], a vector field 8E. Der(Ox) is said to be logarithmic with respect to Dif 8(/) = a] for some aE Ox, where 1is a local (reduced) equation of the germ (D, x) c (X, x). The Ox-module ©A K Peters, ltd. 1058-645812004$ 0.50 perpage Experimental Mathematics 13:4, page 441 442 Experimental Mathematics, Vol. 13(2004), No.4 of logarithmic vector fields (or logarithmic derivations) with respect to Dis denoted by Der( -log D)x and it is closed under the bracket product [-, -]. This yields a 0module coherent sheaf denoted by Der( -log D), which is a submodule of the sheaf of vector fields over X. The divisor Dis said to be free at the point xED (see [Saito 80]) if the Ox-module Der( -log D)x is free (and, in this case, of rank n). The divisor Dis called free if it is free at each point xED. Saito's criterion ([Saito 80]) says that adivisor D == (f == 0) is free at a point xED if and only if there exists abasis {£5 1, ... , £5 n} of Der( -log D)x whose determinant of coefficients with respect to - the partial derivatives is equal to U.: i, for some uEOx such that u(x) =I o. Smooth divisors and normal crossing divisor are free. By [Saito 80], any plane curve DCC2is a free divisor. As stated in [Castro et al. 96], a divisor Dcen is called locally quasi-homogeneous if for each xED there exists a system of local coordinates (Zl, ... ,zn) around xsuch that the germ (D, x) is defined by a weighted homogeneous polynomial h(Zl, ... ,zn) with strictly positive weights for the variables Zi. If a plane curve DCC2 is defined by a weighted homogeneous polynomial, then Dis a locally quasi-homogeneous free divisor-so LCT holds for such a plane curve. In [Calderon et al. 02] a converse of this last result is proved: if Dis a plane curve and LeT holds for D, then Dis locally quasi-homogeneous. It was also shown in [Calderon et al. 02] that, in dimension 3, there are free divisors that verify LCT and are not locally quasi-homogeneous. Adivisor DcX is said to be Euler homogeneous if for each xED there exist a local equation fof the germ I (D, x) and avector field £5 E Der(Ox) vanishing at xsuch that £5(f) == f. In [Calderon et al. 02], it is proved that LeT implies Euler homogeneity for place curves. It is an open problem to describe which free divisors verify LCT in dimension greater than 2, and there is an open conjecture related to this problem: Conjecture 1.1. ([Torrelli 04]) LeT holds for the germ (D, x) if and only if the annihilator ideal Annvx (1/f) is generated by elements of order 1, where fis a local equation of the germ (D,x). Here Vx stands for the ring of germs of linear differential operators with coefficients in the ring Ox and Annv x (1/ f) for the ideal of linear differential operators Pin ti; such that P(I/ f) == o. The order of a nonzero element Pof V x, P== is the integer ord(P) == maxjjo] == a1 +... +a nI ao: =I O} (here ao: E0x and 8i stands for the partial derivative a~i). The principal symbol of Pis by definition the expression a(P) == L ao:~O: , o:,lo:l=ord(P) viewed as an element of the ring of polynomials Ox [~] in n variables, ~ == (~1, ... ,~n), with coefficients in Ox. If V == Vxis the sheaf of rings of linear differential operators with holomorphic coefficients, the stalk of V at xis Vx. The ring Vxis filtered by the order of its elements. The associated graded ring is denoted by gr(V x) . It is easy to prove that gr(V x) is isomorphic to Ox[~]. One of the difficulties in the study of which divisors Dverify LCT is addressed by the examples. Given a divisor Dit is hard to prove whether the inclusion morphism io is a quasl-isomorphism". We propose here a computational tool to test LCT in the free case. An essential ingredient for our method to cover the case of free divisors is a recent result of [Calderon and Narvaez 04]: if Dis a free divisor that verifies LCT then Dis a Spencer divisor. Roughly speaking, Dis a Spencer free divisor (see below) if it is free and admits a special free resolution for the V-module V/ (Der( -log D)) where (Der( -log D)) is the left ideal of V generated by Der( -log D). With the help of this result, it is enough to provide acriterion to test LCT for Spencer-free divisors; the non-Spencer ones do not verify LCT. 2. SPENCER DIVISORS In order to allow an easier reading of this article we review here the results of [Castro and Ucha 02, Sections 3 and 4]. In [Calderon 99] the author associates with a free divisor Dcen the so-called augmented Spencer logarithmic complex V ®o 1\. Der( -log D) -+ M10g D -+ o. It is a complex of V-modules and M10g D stands for the quotient (Der(!'logD)). Definition 2.1. ([Castro and Ucha 02, Definition 3.3]) We say that a free divisor Dis a Spencer divisor if 1 The reader can consider, for example, the ad hoc explicit proof in [Calderon et al. 02] to show that the divisor (xy(x+y)(xz+y) = 0) CC3verifies LCT. Castro-Jimenezand Ucha-Enrfquez: Testingthe Logarithmic Comparison Theorem for FreeDivisors 443 the V-module M10g Dis holonomic and the augmented .Spencer logarithmic complex is a (locally) free resolution of M1ogD. Acoherent V-module Mon X is said to be holonomic if its characteristic variety (see [Mebkhout 89, Chapter I, (2.2)]) has dimension n. Definition 2.2. ([Calderon 99, Definition 4.1.1]) The divisor Dis said to be Koszul free at the point xED if it is free at xand there exists a basis {81, ... ,8n}of Der( -log D)x such that the sequence {O'(8 1) , ••• ,O'(8n)} of principal symbols is a regular sequence in the ring gr(Vx) . The divisor Dis Koszul free if it is Koszul free at any point of D. By [Saito 80] and [Calderon 99, 4.2.2.]- any plane curve DCC2is a Koszul free divisor. By [Calderon 99, Propositions 4.1.2 and 4.1.3] if Dis Koszul free (in particular if Dis a plane curve) then it is a Spencer divisor, but the converse is not true, see [Calderon 99, Remark 4.2.4] and [Castro and Ucha 02, Section 5.3]. Given a coherent V-module M, the solution complexof Mis by definition the complex R'Homv(M,O) {see, for example,. [Mebkhout 89, Chapter I, (2.6)]). This complex will be simply denoted by Sol(M). The following proposition is a consequence of [Calderon 99, Theorem 4.2.1]. Proposition 2.3. If D is a Spencer divisor then there exists anatural quasi-isomorphism from Sol(M1og) to n-(log D). For each xE X, we can consider the ideal Anng~ (1/ f) (here, fis a local equation of the germ (D, x)) generated by the differential operators PEVxsuch that. P(I/ f) =0 and the order of Pis equal to 1. In fact, such an operator Pmust have the form 8+awhere 8is alogarithmic derivation in Der( -log D)x and 8(f) = af with aE Ox' Let us denote by M10g D the quotient V-module with stalks (M1og D) .- Vx x·- Anng~ (1/ f) . The module M10g Dadmits in the Spencer case a free resolution completely analogous to the one of M1og D: V ®o /\- Der(-log D) ~ M10g D ~ 0, where Der( -log D) denotes the free O-module of differential operators that can be written locally as 8+asuch that 8E Der( -log D) and 8(/) = a] for some holomorphic germ a. The following theorem is proved in [Castro and Ucha 02, Theorem 4.3]: Theorem 2.4. For each Spencer divisor DcXwe a have an isomorphism (M1og D)* ~ Mlog D . In this theorem (-)* means duality in the sense of Vmodule theory (see, for example, (Mebkhout89, Chapter I, (4.1) ]). These last two results allow us to use V-module theory in connection with the logarithmic comparison theorem as we will show in the following sections. The following question is open: Problem 2.5. Identify which free divisors are Spencer divisors. An example of a free divisor that is not Spencer is given in [Calderon and Narvaez 04]. Remark 2.6. For a given divisor D == (I =0) with 1ER= C[Xl,"" xn] the modules M10g D and M10g D can be obtained by means of computations of syzygies of polynomials using Grabner bases: if for some mER, then at 8f +...+an 8f - mf = O. aXI aX n Since the inclusion of the ring of differential operators with coefficients in R-the Weyl algebra-in Vxis flat, these computations in Rcover the analytical setting. 3. HOW TO DEDUCETHAT LeT HOLDS Let Dc C" be a free divisor. We state the first criterion to prove that LCT holds. In the proofof its correctness, we use some conditions that are sufficient for the complexes n-(log D) and n-(*D) to be quasi-isomorphic'[. For each germ (D, x) c (C", x) defined by a holomorphic function I, we denote by bl the b-function associated with 1(see [Bernstein 72]). Let us denote by <PD :MlogD ~ O(*D) the natural morphism defined .by <PD(P) =P(-}). For any coherent V-module M the 2The symbol ~ between complexes stands naturally for quasiisomorphisms from now on. 444 Experimental Mathematics, Vol. 13(2004), No.4 de Rham complex DR(M) associated with Mis by definition the complex of sheaves of C-vector spaces V 1 V vM lin 0 o ~ M ~ MQ90 n ~ " ... ~ Q90 ~G ~ , where ni stands for the sheaf of holomorphic differential i-forms on X and \7 is the exterior derivative defined by \7(m Q9 w) == (\7m) 1\ w-(_I)deg(w)m Q9 dw. Criterion 3.1. If Dis a Spencer divisor, then Annv(I/f) == Anng) (I/f) => LCT holds for D. Proof: Let us suppose that Dis Spencer. On the one hand, we have: • By Proposition 2.3, and (see [Mebkhout 89, page 41]) Sol(MlogD) DR((M1ogD)*). • By Theorem 2.4, we deduce that On the other hand, -1 is the smallest integer root of the b-function bf (see [Torrelli 04, Proposition 1.3]), and then we have For the computational aspects, we note: 1. The free divisor D must be Spencer. This condition can be tested using Grabner bases in the Weyl algebra to compute the modules of the syzygies that appear, computing a "free resolution of M10g D and checking whether each module is the one required by the Spencer resolution presented in Definition 2.1. We used the package for V-modules, Macaulay 2 ([Grayson et al. 99]), written by A. Leykin and H. Tsai. 2. The comparison between Annv(I/ f) and Anng) (1/ f) needs the computation of the annihilator. We have used [Noro 02] in the examples. 3. By [Torrelli 04, Proposition 1.3], if Anng) (1/ f) == Annv(I/ f) then the smallest.integer root of the bfunction bfis -1. The global b-function can be computed with the algorithms of [Oaku 97] or [Noro 02]. We have used for the examples the powerful implementation of the latter in [Noro et al. 00]. With respect to the roots of the b-function of free divisors, we can query the following: Problem 3.2. Let D= (f == 0) be a free divisor. Is -1 the smallest integer root of bf? In Figure 1we give examples of free divisors with their global b-functions and their Euler vector fields (i.e., of type X == L Wi Xi 8x i with ui; 2:: 0 and x(f) == c . ffor some c EC \ {O}). Criterion 3.1 applies for all of them, so LeT holds. The examples in our list have been selected because they belong to afamily that we are about to define. where O(*D) is the sheafof meromorphic functions with poles along D and V. -} is its sub-V-module generated (locally) by the meromorphic function 1. So we obtain DR(Ann~(17f)) ~ DR(O(*D)) == ne(*D) as it is clear that V· -} ~ Ann~(l/f). The crucial step is then the comparison between the left ideals Anng) (1/ f) and Annv(I/ f): if Anng) (1/ f) == Annv(I/ f) we obtain that DR(M1ogD) == DR (V ) Anng)(1/ f) = DR (Ann;1/!)) ~ ne(*D). D Definition 3.3. We will say that adivisor DcXis weakly locally quasi-homogeneous if for all xED there are local coordinates (Zl, ... ,zn) on X, centered at x, with respect to which D has adefining equation h(Zl, ... ,zn) E0 such that h(zr 1, ••• ,z~n) is homogeneous of strictly positive weight with the weights ui, positive or (not all) zero. This is the case for our examples in Figure 1. They have weight 0 for the variable Z and the singular locus is the z-axis. It is clear that the change Z ~ Z+Q (QE C) produces a equation that admits the same set of weights. We have found many more examples of weakly locally quasi-homogeneous divisors, and itis apparent that they always verify LCT. We dare to propose the " next conjecture based on this experimental evidence. Castro-Jimenezand Ucha-Enrfquez: Testingthe Logarithmic Comparison Theorem for FreeDivisors 445 (Local) Equation fEuler operator Global b-function bf xy(x + y)(xz +y) x8 x +y8 y(8 + 3/4){8 + 1/2)(8 + 5/4){8 + 1)3 xy(x + y)(x - y)(xz +y) x8 x +y8 y(8 + 1)3(8 + 6/5){8 + 3/5){8 + 2/5)(8 + 4/5) y(x 2+ y)(x 2z +y) x8 x+ 2y8 y(8 + 4/3){8 + 5/6){8 + 1)3 (8 of 1/2){8 + 2/3){8 + 7/6) (xz + y)(x 3_y3) x8 x +y8 y(8 + 3/4){8 + 1/2){8 + 5/4)(8 +1)3 (xz + y)(x 4 _y4) x8 x +y8 y(8 + 1)3(8 + 6/5){8 + 3/5){8 + 2/5){8 + 4/5) (xz + y)(x 7_y7) x8 x +y8 y(8 + 1/2){8 + 7/8){8 + 9/8){8 + 5/8) (8 + 3/4)(8 +1)3(8 + 3/8){8 + 1/4) (8 + 20/17)(8 + 7/17)(8 + 23/17)(8 + 16/17) xy(x 2+y3)(x2z +y3) 3x8 x+ 2y8 y(8 + 13/17){8 + 5/17){8 + 14/17){8 + 12/17) (8 + 10/17)(8 + 18/17){8 + 11/17)(8 + 1)3 (8 + 15/17){8 + 8/17){8 +21/17)(8 + 9/17){8 + 19/17) FIGURE 1. Free divisors that verify LeT. Conjecture 3.4. If D is a weakly locally quasihomogeneous free divisor, then LeT holds. Proving LCT holds for locally quasi-homogeneous free divisors, makes strong use of the fact that such divisors have a structure of an analytical product around any point of the divisor. Our examples do not have this property, so the proofof Conjecture 3.4 has to be more subtle. We think that one of the ways in which the hypotheses of the iocally quasi-homogeneous case could be relaxed is that some of the weights can be zero. The condition on the existence of a local weighted equation seems to be more complex to relax; in the next section we will give examples of free divisors that are defined by weighted homogeneous equations (with all the weights positive) but do not verify LCT. Thus, by [Castro et al. 96], they are not locally quasi-homogeneous (see Remark 5.8). 4. HOW TO DEDUCE THAT LCT DOES NOT HOLD We shall state here a criterion giving a sufficient condition to deduce that LCT does not hold for a given free divisor. This criterion is the converse of Criterion 3.1. Criterion 4.1. If Dis a Spencer divisor, then Annv(l/f) =1= Anng) (l/f) => LCT does not hold for D. Proof: Let us consider the natural sequence of V-module morphism's o----+ K----+ M10g D ~ O( *D) where <PD(P) = P(l) for each PE M10g D and Kis the kernel of <PD, 'We have KAnnv(l/ f) -Anng) (1/ f) · Suppose LCT holds for D. Then the inclusion map io : n-(logD) ~ n-(*D) = DR(O(*D)) is a quasiisomorphism. On the other hand, since Dis Spencer, we have DR(MIOgD)~n-.(logD) (see [Castro and Ucha 02, Theorem 4.3]). Then the composition morphism 446 Experimental Mathematics, Vol. 13(2004), No.4 is a quasi-isomorphism. We have that 'l/J == DR(¢D) by [Calderon and Narvaez 04]; thus, by Proposition 4.2, ¢D must be an isomorphism, contradicting that K i= (0). D Proposition 4.2. [Mebkhout 89, Chapter II, Theorem 4.1.5] Suppose ¢ : M--+ M' is ~ morphism of holonomic V-modules. If DR(¢) : DR(M) --+ DR(M') is a quasiisomorphism, then ¢is an isomorphism. Proof: Taking the kernel and cokernel of ¢, it is enough to show that if Mis a holonomic V-module such that DR(M) == 0, then M == o. So, suppose we have DR(M) == 0 for a holonomic V-module M. By [Mebkhout 89, page 41] we have Sol(M*) ~ DR(M) and then Sol(M*) == o. By applying [Mebkhout 89, Chapter II, Theorem 4.1.5], we get VOO ®v M* == 0 where VOO is the ring of linear differential operators of infinite order. Since VOO is faithfully flat over V(see [Sato et al. 73, Theorem 3.4.1.]), we get M* == 0 and then M == o. D Criteria 3.1 and 4.1 give a necessary and sufficient condition to decide if LCT holds for a Spencer free divisor. In particular, we have proved Conjecture 1.1 for Spencer divisors: Theorem 4.3. Let D be a Spencer free divisor. LeT holds for the germ (D, x) if and only if the annihilator ideal Annv x(1I f) is generated by elements of order 1, where fis a local equation of the germ (D, x). 5. ON THE REGULARITY OF LOGARITHMIC V-MODULES To apply Criterion 4.1 we have chosen an alternative way that, at the same time, treats the following interesting problem: Problem 5.1. Are the logarithmic V-modules M10g D and M10g D regular holonomic for any free divisor D? The answer is yes for plane curves (see [Ucha 99] and [Castro and Ucha 01]) and for all the examples we have studied, including non-Euler homogeneous examples. Moreover, for any Spencer divisor D, since (M1og D)* is isomorphic to M10g D(see Theorem 2.4), the regularity of M10g Dis equivalent to the one of M10g D. The module O(*D) is regular holonomic (see, for example, [Mebkhout 89, Chap. II, Th. 2.2.4]) so it is enough, in order to prove the regularity of M10g D, to prove the same property for the kernel Kof the natural map M10g D ~ O(*D). Proving the regularity of a V-module, computationally, is a very difficult problem in general. We have found many tractable examples due to afriendly presentation of K that we have obtained for our examples in Figure 2. These proposed free divisors are particular cases of the family {Dp,q == ((xz + y)(x p-yq) == 0) CC3}for p, q EN. They are Koszul free'', and therefore Spencer. Criterion 4.1 applies, so they do not verify LCT. It is interesting to point out that every element of the family admits an Euler vector field E == qx8 x +py8 y+ (pq)z8 zE Der( -log Dp,q) with strictly positive weights for all the variables when p>q. This fact means that the defining equation is a weighted homogeneous polynomial. Remark 5.2. It would be interesting to describe the bfunctions for all the elements of the family. The roots seems to follow a pattern related to the weights of the variables, in a way somewhat analogous to the isolated (quasi-homogeneous) singularity case. Let us explain how we have studied the regularity of the kernel Kfor the divisors of the family {Dp,q == (xz + y)(x p-yq) == O)}. To obtain a presentation of the quotient K == Annv(11 f) Anng)(llf) the following procedure is well known: 1. Get a set of generators {9I, , 9r} of Annv (1I f) and a set of generators {lI, , ls} of Anng) (II f)· 2. Compute the V-module Sof syzygies among 91, ... ,9r, lI, ... ,ls· 3. For every generator SE vr+s of S delete its last s components to obtain SED", In this way, if S == (SI, ... ,St) then we have vr K~ __ __; (SI, ... ,St) that is, using a matrix of rrows and t columns (each column is a generator Si). As soon as p and qgrow for Dp,q (for p, q 2: 8), the computations of annihilators, b-functions, and kernels become huge and the examples intractable with the implementation we have used. However, in the tractable cases 3An argument about the dimension of the characteristic variety of M10g Dp,q can be used for the whole family. Castro-Jimenezand Ucha-Enrfquez: Testingthe Logarithmic Comparison Theorem for FreeDivisors 447 (Local) Equation fGlobal b-function bl (xz + y)(x 4_ y3) (8 + 19/15)(8 + 2/3}(8 + 1/2)(8 + 1)3(8 + 13/15) (8 + 17/15)(8 + 4/3)(8 + 16/15)(8 + 14/15)(8 + 5/4) (s + 7/15)(s + 3/4)(8 + 8/15)(s + 11/15) (xz + y)(x 5_ y3) (s + 23/18)(s + 11/18)(s + 19/18){8 + 1/2)(s + 8/9) (8 + 5/4)(s + 5/9)(s + 17/18)(8 + 13/18)(8 +1)3 (s + 25/18)(8 + 3/4){s + 7/9)(s + 11/9)(s + 4/9)(8 + 10/9) (8 + 17/24)(s + 4/3)(8 + 2/3)(s + 23/24){s + 19/24) (XZ + y)(x 7_ y3) (8 + 13/24){s + 5/4){s + 5/12)(8 + 29/24)(s + 25/24) (s + 1)3(s + 7/12)(s + 7/6)(8 + 1/2)(s + 31/24) (s + 5/6)(s + 11/12){s + 35/24)(8 + 3/4)(8 + 13/12) (xz + y)(x 3 .- y4) (s + 13/15)(s + 11/15)(8 + 16/15)(s + 19/15)(8 +1)2 (s + 17/15){s + 2/3)(8 + 4/3)(s + 8/15)(s + 7/15)(8 + 14/15) (XZ + y)(x 3_ y5) (s + 4/9)(8 + 23/18)(s + 25/18)(8 + 13/18)(8 + 11/9) (s + 10/9)(8 + 19/18)(s + 1)2(8 + 17/18) (s + 7/9)(8 + 8/9){8 + 5/9)(s + 11/18) FIGURE 2. Free divisors that do not verify LeT. the kernels turn out to be represented by matrices with a special structure. We have used [Grayson et al. 99] and [Noro et al. 00]4. Example 5.3. In the case p=4 and q=3, the kernel K can be represented (using some elementary simplifications) by the matrix where P= -360;, 48 2 2 3204 2216 Q= -Sz OyOZ +125 zOxoz -25Z0yOZ 17397 417 + '1"25 0xoz -1250y· ( Xy zOz +8 o0 0 o0 0 o0 0 ~Ox 1 o o -1 960 ; P) o0 10' o1 Lemma 5.5. If the V-module Kis represented by the matrix Example 5.4. In the case p=5 and q=3, the kernel can be represented by M= o0 o0 o0 o o o 1 0 o1 o0 o o 1 ( x~ yzoz +5P Q) o0 1 0 , o0 0 1 4More precisely, we have computed Annv(l/ f) computing the annihilator of /8 in 1'[8] with the command AnnFs of [Grayson et al. 99] and replaced sby -1, provided that Risa/ Asir (with the command bf ct) has shown that the smallest integer root of b/ is -1. for any a ECand PI, ... .P; E V, then K ~ V/(x, y, zoz +a). Proof: It is enough. to define the isomorphism cp Vt+ 1/N ~ V/(x,y,zoz +a), 448 Experimental Mathematics, Vol. 13(2004), No.4 where Nis the submodule generated by the columns of M. It is defined as follows: and 81, ... ,8nis the dual basis of Der( -log D), then and iI n-1(V \ 0,Oe n)n ~ iI n-1(V \ 0,Ol (log D )). and its inverse maps I to e1. o The morphism d1 can be read now as Lemma 5.6. If the V-module Kadmits apresentation as in Lemma 5.5, then it is regular holonomic. iI n-1 (V\ 0, Oe n) [g] iI n-1(V \ 0, Oe n)n ([81.g], ... , [8 n.g]) Proof: Obvious from the representation we have obtained for K ~ V/(x, y, z8 z+a), clearly aregular holonomic Vmodule. 0 Theorem 5.7. The elements of the family {Dp,q} with kernels as in Lemma 5.5 do not verify LeT. In addition, the corresponding V-modules M10g Dp ,q and M10g Dp,q are regular holonomic. Proof: Simply apply Criterion 4.1; the corresponding kernels are not null so LCT does not hold. The regularity is a consequence of Lemma 5.6. 0 Remark 5.8. In [Calderon and Narvaez 02, Problem 6.5] the authors ask whether a free divisor defined by a quasihomogeneous polynomial (with strictly positive weights) is locally quasi-homogeneous. The answer to this question is negative: by Theorem 5.7 the first three examples provided in Figure 2 are free divisors defined by quasihomogeneous polynomials (with strictly positive weights) that do not verify LCT. Since locally quasi-homogeneous free divisors verify LCT (see [Castro et al. 96]), these divisors are not locally quasi-homogeneous. We think that every element of the family with p>q(lcm(p, q) =1) are in the same situation, but we don't have a proof for this general result. In [Castro et al. 96] it was noted that if LCT holds then . the morphism is injective,where Vis a Stein neighborhood (sufficiently small) of O. In dimension 2, this condition is equivalent to LCT (see [Calderon et al. 02]). The examples of Figure 2 show that the condition on d1is not sufficient in dimension greater than 2. If {W1,.'.'W n}is a free basis of n1(log D) as Ov-module The space iI n-1(V\O, Oe n)is isomorphic to the space Sof Laurent series, convergent for all ~ = (Xl,".' Xn) with ~ =1= 0 and whose nonzero coefficients are those with strictly negative indices in all variables. It is clear that if there exists an element 81= L~=l WiXi8i with all the ui; 20 in the basis of Der( -log D) it follows that d1is injective, as in the first three examples of Figure 2. 6. CONCLUSIONS Many examples have been treated with the explicit methods we have proposed in this work to study the Logarithmic Comparison Theorem (LCT) on free divisors. The results on the weakly locally quasi-homogeneous examples we have tried has lead us to conjecture that they verify LCT. As a consequence, they would be Spencer divisors. We have proved that LCT holds for a Spencer free divisor D == (f =0) if and only if Annv(I/ f) is generated by differential operators of order 1. On the other hand, we have given examples that answer a question proposed in [Calderon and Narvaez 02]: whether there exist free divisors defined by weighted homogeneous polynomials that are not locally quasihomogeneous. We have proved that for these examples the logarithmic V-modules are regular. ACKNOWLEDGMENTS Both authors were partially supported by BFM-2001-3164 and FQM-333. We are very grateful to Professors L. NarvaezMacarro and Z. Mebkhout for their useful comments. During the preparation of the final version of this work, the first author was visting the Ecole Normale Superieure (Paris). He is grateful to the Departernent de Mathematiques et Applications for its hospitality.