Logarithmic Comparison Theorem and some Euler homogeneous free divisors

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 133, Number 5, Pages 1417–1422 S 0002-9939(04)07678-6 Article electronically published on November 1, 2004 LOGARITHMIC COMPARISON THEOREM AND SOME EULER HOMOGENEOUS FREE DIVISORS F. J. CASTRO-JIM´ ENEZ AND J. M. UCHA-ENR´ IQUEZ (Communicated by Michael Stillman) Abstract. Let D, x be a free divisor germ in a complex manifold Xof dimension n>2. It is an open problem to find out which are the properties required for D, x to satisfy the so-called Logarithmic Comparison Theorem (LCT),that is, when the complex of logarithmic differential forms computes the cohomology of the complement of D, x. We give a family of Euler homogeneous free divisors which, somewhat unexpectedly, does not satisfy the LCT. 1. Introduction It was proved in [2] that if Dis a locally quasi-homogeneous free divisor in the complex manifold X, then the complex Ω•(log D) —introduced in [7]— of holomorphic differential forms with logarithmic poles along Dcalculates the cohomology of the complement of Din X. By analogy with Grothendieck’s Comparison Theorem ([6]) this fact is summarized by saying that this kind of free divisors satisfies the Logarithmic Comparison Theorem (LCT) or simply that LCT holds for D. In [1] a partial converse to the result above was proved: if Dis a reduced plane curve and LCT holds for D,thenDis locally quasi-homogeneous1and it was shown that, in dimension strictly greater than 2, Dcan verify the LCT without being locally quasi-homogeneous.2In [1] it has been conjectured that LCT implies Euler homogeneity. The reciprocal is false: consider the divisor D≡f=(z(x4+y5+xy4)=0)⊂C3. It is Euler homogeneous since (z∂z)(f)=fand is easy to prove that it does not satisfy LCT using that the plane curve defined by x4+y5+xy4= 0 does not either as was proved in [1]. It is an open question which Euler homogeneous free divisors do satisfy LCT. Received by the editors July 21, 2003 and, in revised form, January 8, 2004. 2000 Mathematics Subject Classification. Primary 32S20; Secondary 14F10, 32S40. Key words and phrases. Free divisor, Logarithmic Comparison Theorem, D-modules, Eulerhomogeneous divisor. This work was partially supported by DGESIC BFM-2001-3164 and FQM-333. 1The theorem of K. Saito [7] (for a germ fwith isolated singularity, fbelongs to its Jacobian ideal is equivalent to the quasi-homogeneity of f) is strongly necessary in the proof, so the same argument is no longer valid in higher dimensions. 2It is the case, for example, for D≡(xy(x+y)(xz +y)=0)⊂C3. It is remarkable that this divisor is not analytically a product. c 2004 American Mathematical Society Reverts to public domain 28 years from publication 1417 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1418 F. J. CASTRO-JIM´ ENEZ AND J. M. UCHA-ENR´ IQUEZ Let O=C{x1,...,x n}be the ring of germs of holomorphic functions around 0∈Cnand Dthe ring of differential operators with coefficients in O. Each nonzero element Pin Dcan be uniquely written in the form P=αaα∂αwhere α=(α1,...,α n)∈Nn,aα∈Oand ∂α=∂α1 1···∂αn n(here ∂istands for the partial derivative ∂ ∂xi). The order of Pis by definition the integer number max{|α|= α1+···+αn|aα=0}.A conjecture in this context is the following one: Conjecture 1.1 (in [9]).LCT holds for D⇔the annihilator AnnD(1/h)is generated by elements of order 1,wherehis a local equation of D. This conjecture is true for plane curves (see [10], [4] and [8]). Besides, it has been conjectured that for arrangements of hyperplanes (not necessarily free) LCT holds. The free case is covered in fact in [2] and the case of tame arrangements is covered in [12]. In [11] there are some related questions to 1.1 for the case of central arrangements. So far there are no more general results about the LCT condition. One of the difficulties in the study of this problem is the treatment of the examples. In [3] the authors of this paper proposed computational methods to test LCT, and some partial results —in particular, some experimental backing for 1.1— have been obtained. Nevertheless, the problem for these methods is the limit of the available computer algebra systems for such calculations. In this work we provide a family of Euler homogeneous free divisors that do not satisfy LCT using an entirely different approach. We take advantage of a necessary condition for the LCT that was introduced in [2] and [1]: if LCT holds, then the morphism d1:ˇ Hn−1(V\0,O)→ˇ Hn−1(V\0,Ω1(log D)) is injective, where Vis a Stein neighborhood (sufficiently small) of 0. 2. The injectivity of d1 Given a germ of a divisor D≡(f=0)withf∈Oand following K. Saito [7], the O-module of logarithmic derivations with respect to D,Der(−log D), is the set of derivations δsuch that δ(f)=mf for some m∈O. We make precise here the last paragraph of the introduction. If {ω1,...,ω n} is a free basis of Ω1(log D)asOV-module and δ1,...,δ nis the dual basis of Der(−log D), then ˇ Hn−1(V\0,OCn)≃ˇ Hn−1(V\0,Ω0(log D)) and ˇ Hn−1(V\0,OCn)n≃ˇ Hn−1(V\0,Ω1(log D)). The morphism d1can be read now as ˇ Hn−1(V\0,OCn)d1 →ˇ Hn−1(V\0,OCn)n [g]→ ([δ1·g],...,[δn·g]) since ˇ Hn−1(V\0,OCn) is isomorphic to the space Sof Laurent series, convergent for all x=(x1,...,x n)withx= 0 and whose non-zero coefficients are those with License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LOGARITHMIC COMPARISON THEOREM AND FREE DIVISORS 1419 strictly negative indices in all variables. For p≥nwe consider the following C- vector spaces of finite dimension: Gp= i1,...,in<0 i1+···+in=−p ci1···inxi1 i···xin n, Fp= i1,...,in<0 i1+···+in≥−p ci1···inxi1 i···xin n. Clearly Fp=Gp⊕Gp−1⊕···⊕Gn. The restriction of d1to each Gpis d1|Gp:Gp→(Fp)n, and it is defined as d1|Gp([s]) = (dδ1|Gp([s]),...,d δn|Gp([s])) = ([δ1·s],...,[δn·s]), for all s∈Gp.Ifd1is injective, so is its restriction to each Gp, and this is the core of our method: take successively for each p≥nthe corresponding matrix of d1|Gp and check if it is injective. It is important to point out that if in a basis of Der(−log D)thereexistsa derivation of the form wixi∂iwith wi≥0 and at least one of the wiis nonzero, then the morphism d1|Gpis injective. Although, for plane curves, d1injective and LCT are equivalent conditions (see [1]), this fact is false in higher dimensions: consider again the divisor D≡(z(x4+y5+xy4) = 0) that has the derivation z∂z in a suitable basis of Der(−log D)). 3. A non-Euler homogeneous example In this section we illustrate the use of the injectivity of d1via an example. We study a non-Euler homogeneous divisor (f=0)inC3, that is, for which there doesn’t exist a logarithmic derivation such that δ(f)=f.Itdoesnotsatisfythe LCT and could be considered as a consequence of the behaviour of the Reiffen plane curves (see [4]). Its interest lies in the difficulty of treating this example with the available systems with the methods of [3], due to the size of the Gr¨obner bases computations required. The divisor is D≡(f=(xz +y)(x4+y5+xy4)=0)⊂C3. It is free as a consequence of Saito’s criterion (see [7]) applied to the following basis of Der(−log D): δ1=(xz +y)∂z, δ2=(8x2+10xy)∂x+(6xy +8y2)∂y+(−2yz +2y)∂z, δ3=(xy2+1 4y3+5 4x2−25 4xy)∂x +(3 4y3−1 4x2+5 4xy −5y2)∂y +(1 4y2z2−1 4y2z+5 4yz +1 4x)∂z, whose determinant (of coefficients with respect to ∂1,∂ 2,∂ 3)is−2·f.Itisnot Euler homogeneous since the ideal generated by the aiwhere δi(f)=aif(where a1=x, a2=40x+48y,a3=1 4y2z+19 4y2+25 4x−30y) is contained in the ideal generated by x, y, z. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1420 F. J. CASTRO-JIM´ ENEZ AND J. M. UCHA-ENR´ IQUEZ We show that d1is not injective, because it is not injective on G3.Theonly element we have to check is 1 xyz .Wehave δ11 xyz =(xz +y)−1 xyz2=0, δ21 xyz =(8x2+10xy)−1 x2yz +(6xy +8y2)−1 xy2z +(−2yz +2y)−1 xyz2=0, δ31 xyz =(xy2+1 4y3+5 4x2−25 4xy)−1 x2yz  +(3 4y3−1 4x2+5 4xy −5y2)−1 xy2z +(1 4y2z2−1 4y2z+5 4yz +1 4x)−1 xyz2=0. This study can be extended (although it is a little tedious!) to the family of divisors Dp,q ≡((xz +y)(xp+yq+xyq−1)=0)⊂C3for 4 ≤p<qor even in any dimension considering Dp,q ≡((xp 1+xq 2+x1xq−1 2)(n i=3(x1xi+x2)) = 0),4≤p<q, as we will do in the next section for a more interesting family. 4. A family with “Euler vector fields” Here we present a family of Euler homogeneous free divisors that do not satisfy the LCT. In our opinion it is an interesting family because: •They have not been constructed starting with non-quasi-homogeneous plane curves but from quasi-homogeneous cuspids. •They have in their bases of logarithmic derivations an Euler vector field in a broad sense of the word: there exists a vector field E=n i=1 wixi∂i,w i∈ Q,withE(f)=f,wheref= 0 is a local equation of the divisor. So let us consider the family Fof divisors Dp,q ≡fp,q =(xp 1−xq 2) n  i=3 (x1xi+x2)=0 ⊂Cn with p<q. We will prove that •Every element of Fis an Euler homogeneous free divisor. •For a (infinite) subfamily Eof F,d1is not injective, so LCT does not hold. Proposition 4.1. The elements of Fare (Euler homogeneous) free divisors. Proof. By Saito’s criterion, [7], it is enough to show a collection of nelements in Der(−log Dp,q) whose determinant (of the coefficients of the derivations ∂1,...,∂ n) is equal to u·fp,q where uis a polynomial such that u(0) =0. Wetake δ1=qx1∂1+px2∂2+n i=3(p−q)xi∂i, δ2=q p(q−p)x1xq−2 2xn+1 pxq−1 2∂1 +1 q−pxq−1 2xn+1 qxp−1 1∂2 +n−1 i=3 1 pxq−2 2x2 i−1 pxq−2 2xixn−1 qxp−2 1∂i−1 qxp−2 1∂n, δi=(x1xi+x2)∂ifor i=3,...,n. The corresponding determinant is equal to fp,q. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LOGARITHMIC COMPARISON THEOREM AND FREE DIVISORS 1421 Note that δ1is an Euler vector field with negative weights for x3,...,x nand positive for x1,x 2. Theorem 4.2. If q+p+n i=3(p−q)αi=0,αi>0and 3≤p<q,then d11 x1x2xα3 3···xαn n=0in ˇ Hn−1(V\0,OCn)n. So d1is not injective and LCT does not hold. Proof. First, it is clear that δ11 x1x2xα3 3···xαn n=(q+p+ n  i=3 (p−q)αi)1 x1x2xα3 3···xαn n =0 by the diophantine hypothesis on p, q and the αi. Second we have δi1 x1x2xα3 3···xαi i···xαn n =−αix1xi x1x2xα3 3···xαi+1 i···xαn n +x2 x1x2xα3+1 3···xαn n=0, for i=3,...,n. Finally, δ21 x1x2xα3 3···xαn n =q p(q−p)x1xq−2 2xn+1 pxq−1 2−1 x2 1x2xα3 3···xαn n +1 q−pxq−1 2xn+1 qxp−1 1−1 x1x2 2xα3 3···xαn n +n−1 i=3 1 pxq−2 2x2 i−1 pxq−2 2xixn−1 qxp−2 1−1 x1x2xα3 3···xαi+1 i···xαn n −1 pxp−2 1−1 x1x2xα3 3···xαn−1 n−1xαn+1 n =0. The key is again that we have eliminated x1or x2of every Laurent monomial so that it becomes 0 in ˇ Hn−1(V−0,OCn). Remark 4.3.We have also studied the case p=2<q, that is, the family of divisors D2,q ≡(f2,q =(x2 1−xq 2)n i=3(x1xi+x2)=0)⊂Cnfor many values of q,and they satisfy LCT: it is apparent that this fact is true for any q>2, but we do not have any proof of this result. In all the examples, we have obtained that the annihilator ideal AnnD(1/f2,q) is generated by elements of order 1, and then we use the Duality Theorem of [5]. We would like to point out that for p=2<qthe corresponding divisor is a product (analytically): it is a consequence of the presence of the derivation δ2= ···− 1 2∂n. A different approach to prove LCT for D2,q,forn= 3, could be to find an analytical change of variables producing a weighted homogeneous equation in 2 variables. 5. Conclusions We have provided a family of Euler homogeneous free divisors (in any dimension greater than 2) —defined by polynomials that are “weakly weighted homogeneous” with some negative weights— that do not satisfy LCT. It is an interesting question whether this set of weights is valid locally around 0 for this family, that is, if for License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1422 F. J. CASTRO-JIM´ ENEZ AND J. M. UCHA-ENR´ IQUEZ any point xin a small enough neighborhood of 0 the divisor (D,x) admits a local weighted homogeneous equation hwith eventually some negative weights. Acknowledgements We deeply thank Prof. David Mond for all his suggestions and patience during the elaboration of this work. 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MR1407505 (97h:52013) Departamento de ´ Algebra, Facultad de Matem´ aticas, Universidad de Sevilla, Apdo 1160, E-41080 Sevilla, Spain E-mail address:[email protected] Departamento de ´ Algebra, Facultad de Matem´ aticas, Universidad de Sevilla, Apdo 1160, E-41080 Sevilla, Spain E-mail address:[email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use