Linearity conditions on the Jacobian ideal and logarithmic--meromorphic comparison for free divisors

arXiv:0804.2219v1 [math.AG] 14 Apr 2008 Linearity conditions on the Jacobian ideal and logarithmic–meromorphic comparison for free divisors L. Narv´aez Macarro∗ Dedicated to Lˆe D˜ung Tr´ang on his sixtieth birthday Abstract In this paper we survey the role of D-module theory in the comparison between logarithmic and meromorphic de Rham complexes of integrable logarithmic connections with respect to free divisors, and we present some new linearity conditions on the Jacobian ideal which arise in this setting. MSC: 32C38; 14F40; 32S40 Key words: Free divisor, Jacobian ideal, logarithmic forms, D-module, Bernstein polynomial Introduction The comparison between the meromorphic and the logarithmic de Rham complexes of a vector bundle endowed with a logarithmic integrable connection originally appeared in the case of normal crossing divisors [14]. In [31], the notion of free divisor was introduced, and it produced reasonable logarithmic de Rham complexes with applications to Singularity theory. In [10], a logarithmic-meromorphic comparison (for the trivial bundle) was proved for locally quasi-homogeneous free divisors. The proof is topological and is based on Grothendieck’s comparison theorem. In the appendix A of [15], for the normal crossing case, and in [3], for general free divisors, D-module theory enters the scene and the bases for an algebraic treatment of the logarithmic-meromorphic comparison (at least for free divisors) were established. D-module theory has been used for characterizing the logarithmic-meromor- phic comparison for the trivial bundle with respect to a free divisor in [12], [36], [6], and in [6], [8] for general integrable connections, and also for proving algebraically the logarithmic-meromorphic comparison under the additional purely ∗Partially supported by MTM2004-07203-C02-01, MTM2007-66929 and FEDER. 1 algebraic hypothesis that the Jacobian ideal is of linear type. This hypothesis had been previously proved for locally quasi-homogeneous free divisors in [5]. In this paper we survey the above results and we highlight some (we believe) new linearity type conditions which appear in many examples and which deserve further study. These conditions seem to play a role in understanding algebraically the logarithmic-meromorphic comparison, and also have interesting interactions with Bernstein polynomials and Bernstein functional equations. Let us now comment on the content of this paper. Section 1 is a survey on the use of D-module theory in the logarithmicmeromorphic comparison and on the previous results. In section 2 we first recall the notions of “commutative linear type” and “differential linear type”. We also give an algebraic criterion for checking the (pre)Spencer property. Second, we introduce some generalized linear type properties, which are illustrated in the examples in the third section, and we study their relationship with “classical” properties. We end the section with a list of open questions. In section 3 we have collected several examples. As the reader can guess, the computational complexity is often hard and we comment on the strategies we have followed. We thank F.J. Calder´on Moreno, F.J. Castro Jim´enez and T. Torelli for many discussions on this subject. We also thank J. Mart´ın Morales for computational help. Finally, we thank the referee for his careful reading of our paper and for his comments. 1 A survey on the logarithmic comparison problem for free divisors 1.1 Notations and basic notions Let Xbe a n-dimensional complex analytic manifold and D⊂Xa hypersurface (= divisor), and let us denote by j:U=X−D ֒→Xthe corresponding open inclusion. We denote by OXthe sheaf of holomorphic functions on X, ID⊂OXthe ideal of D,DXthe sheaf of linear differential operators on X(with holomorphic coefficients), Gr DXthe graded ring associated with the filtration Fby the order and σ(P) the principal symbol of a differential operator P. If J⊂DXis a left ideal, we denote by σ(J) the corresponding graded ideal of Gr DX. We denote by ωXthe sheaf of n-differential forms on X, which carries a canonical structure of right DX-module. Given a complex of left DX-modules M, we will denote by DR Mits de Rham complex: DR M=RHomDX(OX,M) = ωX L ⊗DXM[−n]. If Mis a single left DX-module, the above definition coincides with the classical one DR M=M→Ω1 X⊗OXM→ · · · → Ωn X⊗OXM. 2 Let us denote by Jac(D)⊂OXthe Jacobian ideal of D⊂X, i.e. the coherent ideal of OXwhose stalk at any p∈Xis the ideal generated by h, ∂h ∂x1,..., ∂h ∂xn, where h∈OX,p is any reduced local equation of Dat pand x1,...,xn∈OX,p is a system of local coordinates centered at p1. For any bounded complex Kof sheaves of C-vector spaces on X, let us denote by K∨=RHomCX(K,CX) its Verdier dual. If Ais a commutative ring (or a sheaf of commutative rings) and Man A-module, we will denote by SymA(M) its symmetric algebra. If I⊂Ais an ideal, we will denote by R(I) = ⊕∞ d=0Idtd⊂A[t] its Rees algebra. The sheaf of meromorphic functions along Dis denoted by OX(⋆D). It is filtered by the invertible sheaves OX(rD), r≥0, formed by meromorphic functions with poles along Dof order at most r. As usual, let us also denote by OX(−rD) de ideal of OXof functions vanishing at Dwith order al least r,r≥0. We have OX(0D) = OX,OX(−D) = ID,OX(rD)⊗OXOX(sD) = OX((r+s)D) and OX(−rD) = OX(rD)∗for r, s ∈Z. A key result in D-module theory is that OX(⋆D) is a left holonomic DX-module. Let us denote by Ωt X(⋆D) the sheaf of meromorphic t-forms with poles along D, for t= 0, . . . , n, and Ω• X(⋆D) the meromorphic de Rham complex (along D). We say that a meromorphic t-form (with poles along D)ωis logarithmic if both ωand dω have simple poles along D. The sheaf of logarithmic t-forms along Dis denoted by Ωt X(log D). By definition, logarithmic forms along Dendowed with the exterior differential give rise to a subcomplex of Ω• X(⋆D), which is called logarithmic de Rham complex (along D)and is denoted by Ω• X(log D). The OX-module of (holomorphic) vector fields on X, or equivalently, the C- derivations of OX, will be denoted by DerC(OX). A (holomorphic) vector field δon Xis called logarithmic (along D)if the ideal of Dis fixed by δ, i.e. if it is tangent to Dat the smooth locus. The sheaf of logarithmic vector fields (along D) is denoted by Der(log D). The bracket of two logarithmic vector fields is still logarithmic. Let x1,...,xnbe a system of local coordinates on a open neighborhood Uof a point p∈Dand let hbe a reduced local equation of Don U. The module of syzygies (a0, a1,...,an) of (h, h′ x1,...,h′ xn) is isomorphic to the sheaf of logarithmic vector fields on Uthrough the correspondences (a0, a1,...,an)99K n X i=1 ai ∂ ∂xi , −δ(h) h, δ(x1),...,δ(xn)L99 δ and so Der(log D) is a coherent module. The above construction can be performed without choosing local coordinates by considering the map P∈F1DX7→ P(h)∈Jac(D) (1) 1This definition of Jacobian ideal is independent of the (reduced) equation hand does not match the usual definition in Singularity Theory, where the Jacobian ideal is generated by ∂h ∂x1,..., ∂h ∂xnand depends on h. 3 provided that Dhas a reduced global equation h:X→C. The usual pairing of vector fields and differential forms establishes a perfect OX-duality between Der(log D) and Ω1 X(log D) and then both modules are reflexive [31], cor. (1.7). We say that Dis a free divisor [31] if the OX-module Der(log D) is locally free (necessarily of rank n), or equivalently if the OX-module Ω1 X(log D) is locally free (of rank n). Normal crossing divisors, plane curves, free hyperplane arrangements (e.g. the union of reflecting hyperplanes of a complex reflection group), discriminants of stable mappings or bifurcation sets of holomorphic functions are examples of free divisors. More recently, another interesting source of examples of free divisors has been studied in [17], [2]. Is Dis a free divisor, then for any t≥1 we have Ωt X(log D) = VtΩ1 X(log D) [31]. The logarithmic-meromorphic comparison problem for Dconsists of whether the inclusion Ω• X(log D)֒→Ω• X(⋆D) is a quasi-isomorphism or not, or equivalently by Grothendieck’s comparison theorem [19], whether the canonical morphism Ω• X(log D)→Rj∗CUis an isomorphism (in the derived category) or not. The meromorphic de Rham complex can be described in terms of D-module theory as Ω• X(⋆D) = DR OX(⋆D). In sections 1.6 and 1.7 we will explain how, at least in the case of free divisors, the logarithmic de Rham complex can be also described in terms of D-module theory (see corollary (1.6.2)) and how the logarithmic-meromorphic comparison problem can be translated into a comparison between certain DX-modules (see theorem (1.7.1)). 1.2 Lie algebroids The module DerC(OX) of vector fields on Xis at the same time an OX-module and a sheaf of C-Lie algebras. The interplay between both structures can be summarized by the equality [δ, fδ′] = f[δ, δ′] + δ(f)δ′ for any vector fields δ, δ′and any holomorphic function f. In order to understand how logarithmic differential operators (see section 1.3) and logarithmic vector fields are related, at least in the case of free divisors, and the formal analogies with vector fields and differential operators, it is useful to consider the notion of Lie algebroid. A general reference for this notion is [22]. See also [13]. (1.2.1) Definition. A Lie algebroid (on X) is an OX-module Lendowed with a structure of sheaf of C-Lie algebras and a OX-linear morphism ρ:L→ DerC(OX), called anchor morphism, which is also a morphism of sheaves of Lie algebras and satisfies [λ, fλ′] = f[λ, λ′] + ρ(λ)(f)λ′for any local sections λ, λ′ of Land any holomorphic function f. The notion of morphism of Lie algebroids is clear and is left up to the reader. 4 (1.2.2) Example. 1) The first example of Lie algebroid is L=DerC(OX) with the identity as anchor morphism. 2) The sheaf of differential operators of degree ≤1, F1DX=OX⊕DerC(OX), with the projection F1DX→DerC(OX) as anchor morphism, is a Lie algebroid. 3) Any submodule L⊂DerC(OX) which is closed for the bracket is a Lie algebroid with the inclusion as anchor morphism. This applies in particular to L=Der(log D). An OX-ring is a sheaf of rings Ron Xendowed with a morphism of sheaves of rings OX→R. We say that the OX-ring Ris central over Cif the map OX→Rsends the constant sheaf CXinto the center of R. For instance, the sheaf DXof linear differential operators is an OX-ring central over C. (1.2.3) Definition. If (L, ρ) is a Lie algebroid on Xand Ris an OX-ring central over C, we say that a C-linear map ϕ:L→Ris admissible if the following properties hold: (a) ϕis left OX-linear. (b) ϕ([λ, λ′]) = ϕ(λ)ϕ(λ′)−ϕ(λ′)ϕ(λ) for any local sections λ, λ′of L. (c) ϕ(λ)f=fϕ(λ) + ρ(λ)(f)1Rfor any local section λof Land any holomorphic function f. Any Lie algebroid Lhas a functorially associated enveloping algebra U L (cf. [30] and [7], th. (1.6)), which is an OX-ring central over C, with a universal admissible map L→UL. The enveloping algebra U Lis filtered with F0UL= OX,F1UL=OX⊕Land its graded algebra is commutative. We have then a canonical morphism of commutative OX-algebras SymOXL→Gr U L.(2) (1.2.4) Example. 1) In the case of L=DerC(OX) the enveloping algebra is nothing but the ring DXof differential operators with the filtration by the order. 2) In the case of L=F1DXthe enveloping algebra is the polynomial ring DX[s], where sis a central variable, endowed with the total order filtration FTgiven by Fk TDX[s] = k X i=0 (FiDX)sk−i,∀k≥0. The universal admissible morphism F1DX=OX⊕DerC(OX)→DX[s] is the identity on DerC(OX) and sends any holomorphic function fto fs. 3) The case L=Der(log D) will be treated in the following section when Dis a free divisor. We have the following version of the Poincar´e-Birkhoff-Witt theorem. Its proof can be easily deduced from the global case of Lie-Rinehart algebras [30], th. 3.1. 5 (1.2.5) Theorem. If Lis a Lie algebroid on Xwhich is locally free of finite rank as OX-module, then the canonical morphism (2) is an isomorphism. (1.2.6) Let us assume that Lis a Lie algebroid on Xwhich is locally free of finite rank as OX-module and let Ebe a left U L-module. The Cartan- Eilenberg-Chevalley-Rinehart-Spencer complex SpLEis defined in [8], (1.1.7). It is a complex of left U L-modules and carries an obvious natural augmentation ε0: Sp0 LE= U L⊗OXE→h0(SpLE) = E. One easily sees ([8], prop. (1.1.8)) that if the left U L-module Eis locally free of finite rank over OXthen SpLEis a U L-locally free resolution of the UL-module E, that will be simply called Spencer resolution of E. 1.3 Logarithmic differential operators The Malgrange-Kashiwara V-filtration with respect to Don the sheaf DXof linear differential operators on Xis defined by2 VD kDX={P∈DX|P(Im D)⊂ Im−k D∀m∈Z}, k ∈Z. It is an increasing filtration and VD 0DXis a subsheaf of rings of DXwith F0VD 0DX=OXand F1VD 0DX=OX⊕Der(log D). We have then a canonical morphism of commutative OX-algebras SymOXDer(log D)→GrFVD 0DX(3) and a canonical morphism of non commutative OX-rings UDer(log D)→ VD 0DX.(4) The 0’th term of the Malgrange-Kashiwara filtration with respect to Don the sheaf DX,VD 0DX, is also denoted by DX(log D), or simply by V0if no confusion is possible, and its sections are called logarithmic differential operators with respect to D. The following theorem is not explicitly stated in [3], but it is essentially contained in proposition 2.2.5 of loc. cit. (see also [33]). (1.3.1) Theorem. If Dis a free divisor, then the canonical morphism (4) is an isomorphism. The proof of the above theorem given in [3] is based on the corollary 2.1.6 of loc. cit., which states that if Dis free the morphism (3) is an isomorphism of commutative OX-algebras. Let us note that, for proving corollary 2.1.6, it is possible to avoid theorem 2.1.4 and to obtain it directly from proposition 2.1.2 in [3]. As a consequence of theorem (1.3.1) we deduce that if Dis free, then V0is coherent and it has noetherian stalks of finite global homological dimension. If 2It was originally defined only for smooth D. 6 {δ1,...,δn}is a local basis of the logarithmic vector fields on a connected open set V, any differential operator in Γ(V, V0) can be written locally in a unique way as a finite sum X α∈Nn |α|≤d aαδα1 1···δαn n,(5) where the aαare holomorphic functions on V. 1.4 Integrable logarithmic connections with respect to a free divisor Let Ebe a OX-module. A logarithmic connection with respect to Don Eis aC-linear morphism ∇:E−→ Ω1 X(log D)⊗OXEsatisfying Leibniz’s rule, i.e. ∇(ae) = a∇(e) + da ⊗efor any holomorphic function aand any section eof E. The data of a such logarithmic connection is equivalent to the data of a left OX-linear map ∇•:Der(log D)−→ EndC(E), where the Leibniz rule becomes ∇δ(ae) = a∇δ(e) + δ(a)efor any logarithmic vector field δ, any holomorphic function aand any section eof E. From now on let us assume that Dis a free divisor and that ∇is a logarithmic connection on E. As in the classical case (cf. [14], chap. I, 2.10) we define a sequence of C-linear morphisms E∇ −→ Ω1 X(log D)⊗E∇ −→ Ω2 X(log D)⊗E∇ −→ · · · ∇ −→ Ωn X(log D)⊗E.(6) We say that ∇is integrable if the sequence (6) is a complex of sheaves of complex vector spaces. As in the classical case cf. [14, chap. I, 2.12–2.14], ∇ is integrable if and only if the morphism ∇•preserves Lie brackets (i.e. it is admissible in the sense of definition (1.2.3) ), or merely if ∇2:E→Ω2 X(log D)⊗E vanishes. If ∇is integrable then the complex (6) will be called the logarithmic de Rham complex of E(endowed with ∇) and will be denoted by Ω• X(log D)(E). The following proposition is a straightforward consequence of theorem (1.3.1) (see [3], cor. 2.2.6) and it is similar to the well known case of integrable connections and left DX-module structures. (1.4.1) Proposition. To give an integrable logarithmic connection on Eis equivalent to giving a structure of left DX(log D)-module on Eextending its structure of OX-module. The following theorem is the first step in describing logarithmic de Rham complexes in terms of D-module theory (see corollary (1.6.2)). It is a consequence of theorem (1.3.1) and of the existence of the Spencer resolution of OX (1.2.6). (1.4.2) Theorem. ([3], cor. 3.2.2) For any left V0-module Ethere is a canonical isomorphism in the derived category RHomV0(OX,E)≃Ω• X(log D)(E). 7 From now on we will only consider integrable logarithmic connections on OX- modules which are locally free of finite rank, and such an OX-module endowed with an integrable logarithmic connection (with respect to D) will be simply called an ILC (with respect to D). In other words, an ILC will be a left V0- module which is locally free of finite rank over OX. The first examples of ILC are the invertible OX-modules OX(mD)⊂OX(⋆D), m∈Z. If f= 0 is a reduced local equation of Dat p∈Dand δ1,...,δnis a local basis of Der(log D)pwith δi(f) = αif, then f−mis a local basis of OX,p(mD) over OX,p and from (5) we have the following local presentation over DX,p(log D) OX,p(mD)≃DX,p(log D)/DX,p(log D)(δ1+mα1,...,δn+mαn). For E=OXwe have Ω• X(log D)(OX) = Ω• X(log D). For any ILC Eand any integer m, the locally free OX-modules E(mD) := E⊗OXOX(mD) and E∗:= HomOX(E,OX) are endowed with a natural structure of left V0-module (cf. [6], §2 and [7], §2), and they are again ILC, and the usual isomorphisms E(mD)(m′D)≃E((m+m′)D), E(mD)∗≃E∗(−mD) are V0- linear. For any ILC Ethe complex Sp E:= SpDer (log D)Eis a resolution of Eas left V0-module. In particular, any ILC is coherent as left V0-module. 1.5 Koszul and Spencer properties In this section we recall two natural properties which appear when studying free divisors and the logarithmic-meromorphic comparison problem. From now on D⊂Xis assumed to be a free divisor. (1.5.1) Definition. ([3], def. 4.1.1) The (free) divisor Dis said to be Koszul at a point p∈Dif the symbols of any (or some) local basis {δ1,...,δn}of Der(log D)pform a regular sequence in Gr DX,p. We say that Dis a Koszul divisor if it is so at any point p∈D. For the free divisor D, to be Koszul is equivalent to being holonomic in the sense of [31], def. (3.8), i.e. the logarithmic stratification of Dis locally finite3. Any plane curve is a Koszul free divisor (see the proof of corollary 4.2.2 in [3]). Any locally quasi-homogeneous4free divisor Dis Koszul [5]. Examples of locally quasi-homogeneous free divisors are free hyperplane arrangements or discriminants of stable maps in Mather’s “nice dimensions”. (1.5.2) Definition. ([11], def. 3.3) The (free) divisor Dis said to be Spencer if the complex of DX-modules DX⊗V0Sp OXis holonomic and concentrated in degree 0. 3This was noticed by M. Schulze. 4i.e. for which for any p∈Dthere is a system of local coordinates xcentered at psuch that the germ (D, p) has a reduced weighted homogeneous defining equation (with strictly positive weights) with respect to x. 8 Since Sp OXis a locally free resolution of the Der(log D)-module OX, we have DX L ⊗V0OX=DX⊗V0Sp OXand so Dis Spencer if and only if hiDX L ⊗V0OX= 0 for i6= 0 and h0DX L ⊗V0OX=DX⊗V0OX=DX/DXDer(log D) is holonomic. (1.5.3) Definition. The (free) divisor Dis said to be pre-Spencer if the complex of DX-modules DX⊗V0Sp OXis concentrated in degree 0. We have the following result ([6], prop. 1.2.3). (1.5.4) Proposition. Any Koszul free divisor is Spencer. There are lots of examples of Spencer free divisors which are not Koszul (see for instance the examples in section 3). It should be noted that (pre)Spencer property is not easy to check algebraically and in principle one needs to check that each hi(DX⊗V0Sp OX) vanishes, for i= 1, . . . , n. However, see proposition (2.2.17), corollary (2.2.18) and corollary (1.6.4). 1.6 Duality In this section we review some results of [6] on the relationship between the duality of integrable logarithmic connections with respect to a free divisor D⊂ Xand the duality of D-modules. Let us denote by DDX:Db coh(DX)→Db coh(DX) the duality functor of D-module theory (cf. [24], def. I.4.1.6). It is defined by DDXM=HomOX(ωX, R HomDX(M,DX))[n]. The duality functor DDXis a contravariant involutive self-equivalence of the derived category Db coh(DX) of bounded complexes of left DX-modules with coherent homologies, and it induces a contravariant involutive self-equivalence of the abelian category of left holonomic DX-modules. If we start with an ILC (with respect to D)E, we can take first its dual (as ILC) and second the scalar extension from V0to DX, and we obtain DX L ⊗V0E∗, or we can reverse the order and take first its scalar extension and second its dual (as DX-module), and we obtain DDX(DX L ⊗V0E). How are both results related? The following theorem gives a precise answer to this question. (1.6.1) Theorem. ([6], cor. 3.1.2) Let Ebe an ILC (with respect to D). There is a natural isomorphism in Db coh(DX): DDXDX L ⊗V0E≃DX L ⊗V0E∗(D). 9 local basis at pof the logarithmic vector fields (with respect to D), that we can take in such a way that δi(f) = 0 for i= 1,...,n−1and δn(f) = f. The following properties are equivalent: (a) Dsatisfies property (GK)pat p. (b) σ(δ1),...,σ(δn−1)is a regular sequence in GrFD. Proof. Let x1, . . . , xnbe a system of coordinates centered at pand let us write ξi=σ(∂/∂xi) and σi=σ(δi). We have GrFTD[s] = O[s, ξ]⊃GrFD=O[ξ]. It is clear by faithful flatness that σ1,...,σn−1, σn−sis a regular sequence in O[s, ξ] if and only if σ1,...,σn−1is a regular sequence O[ξ].  The following proposition (with its corollary) gives a criterion to prove that a free divisor is pre-Spencer. (2.2.17) Proposition. Let us assume that Dis given by a global reduced equation f:X→Cand that the complex DX[s]⊗U Θf,s SpΘf,s OXis concentrated in degree 0, i.e. it is a DX[s]-resolution of DX[s]/DX[s]·Θf,s (see (8)). Then, the following properties are equivalent: (a) The complex DX⊗V0Sp OXis exact in degree −1. (b) The module DX[s]/DX[s]·Θf,s has no s-torsion. (c) The divisor Dis pre-Spencer. Proof. To prove the proposition we can proceed locally at any point p∈D. Let us write O=OX,p and D=DX,p, and let {δ1,...,δn}be a O-basis of Der(log D)p, and {δ1−α1s, . . . , δn−αns}the corresponding basis of (Θf,s)p, with δi(f) = αif. Let us write, for i < j, [δi, δj] = Pn k=0 aij kδkand Rij = (aij 1,..., i z}| { δj+aij i,..., j z}| { −δi+aij j,...,aij n) the corresponding syzygy of δ1,...,δn. Let us also write Rij (s) = (aij 1,..., i z}| { δj−αjs+aij i,..., j z }| { −δi+αis+aij j,...,aij n) the corresponding syzygy of δ1−α1s,...,δn−αns. We have Rij (0) = Rij . (a) ⇒(b) Let P(s)∈D[s] such that sP(s)∈D[s]·Θf,s: sP(s) = n X i=1 Ai(s) (δi−αis). 16 By taking s= 0, we obtain 0 = Pn i=1 Ai(0)δiand from (a) we deduce that (A1(0),...,An(0)) = X i<j cijRij . Let us consider (B1(s),...,Bn(s)) = (A1(s),...,An(s)) −X i<j cijRij (s). By taking s= 0 again we obtain that all the Bi(0) vanish, so Bi(s) = sB′ i(s), sP(s) = n X i=1 Ai(s) (δi−αis) = n X i=1 Bi(s) (δi−αis) = s n X i=1 B′ i(s) (δi−αis) and P(s)∈D[s]·Θf,s. (b) ⇒(c) Let Cbe the stalk at pof DX[s]⊗U Θf,s SpΘf,s OX. It is a free resolution of M:= D[s]/D[s]·Θf,s. The complex C′:= DL ⊗D[s]Mcan be computed through the D[s]-free resolution of Dgiven by D[s]s· −→ D[s] and so, since Mhas no s-torsion, the complex C′=DL ⊗D[s]M=−1 Ms· −→ 0 M is concentrated in degree 0. But C′can be also computed as C′=D⊗D[s]C and the second complex is nothing but the stalk at pof DX⊗V0Sp OX, and so it is concentrated in degree 0. (c) ⇒(a) It is trivial.  (2.2.18) Corollary. Let us assume that Dis given by a global reduced equation f:X→Cand that Dsatisfies property (GK). Then, the following properties are equivalent: (a) The complex DX⊗V0Sp OXis exact in degree −1. (b) The module DX[s]/DX[s]·Θf,s has no s-torsion. (c) The divisor Dis pre-Spencer. Proof. It is enough to prove that the property (GK) implies that the complex L=DX[s]⊗U Θf,s SpΘf,s OXis concentrated in degree 0, but this is easily proved by filtering Lin such a way that its graded complex is the Koszul complex associated with the sequence δ1−δ1(f) fs, . . . , δn−δn(f) fs 17 over GrFTDX[s] = SymOXF1DX, where {δ1,...,δn}is a local basis of the logarithmic vector fields (with respect to D). See [8], prop. (1.5.3) for the details.  (2.2.19) Example. Corollary (2.2.18) is applied in examples 3.1, 3.3.1, 3.3.2 to deduce the pre-Spencer property. However, it does not apply to example 3.2. In [37] the reader can find some related linearity properties and their relationship with the LCT. (2.2.20) Remark. The property of being pre-Spencer is a non-commutative version of the notion of regular sequence for δ1,...,δn. However, in the noncommutative case the pre-Spencer property is not inherited by subsequences, when that makes sense (see example 3.2). 2.3 Further linearity conditions: examples and questions For a divisor D⊂X, the properties of being of commutative linear type or of differential linear type are very restrictive. In dimension 2 we have the following proposition. (2.3.1) Proposition. Let assume that dim X= 2, and so Dis automatically free. The following properties are equivalent: (a) Dis of commutative linear type. (b) Dis of differential linear type. (c) Dsatisfies the LCT. (d) Dis Euler homogeneous. (e) Dis locally quasi-homogeneous. Proof. The implication (a) ⇒(b) comes from proposition (2.2.9). The equivalences (c) ⇔(d) ⇔(e) have been proved in [4]. The implication (e) ⇒(a) is a consequence of theorem (2.2.7). For the remaining implication (b) ⇒(d), we use some results of T. Torrelli (see [8], remark (1.6.6), c)). We can proceed locally and assume that Dhas a reduced equation f= 0. Since dim X= 2, we know that Dis free and Koszul. Let {δ1, δ2}be a local basis of the logarithmic derivations, with δi(f) = αif. Property (b) means that annD[s]fsis generated by δ1−α1s, δ2−α2s. But the Bernstein polynomial of fhas no integer roots less than −1 (cf. [38]), and so annDf−1is generated by δ1+α1, δ2+α2. We can now apply proposition 4.1 and lemma 4.3 of [36] to conclude that fbelongs to the ideal generated by its partial derivatives, i.e. Dis Euler homogeneous.  In higher dimension, the relationship between the properties in the above proposition, even for free divisors, is not clear (see conjecture 1.4 in [4] and 18 [16]). For instance, examples in Section 3 satisfy the LCT but they are not of differential linear type. Nevertheless, some generalized linear type conditions appear. Let us denote by DX(s) := C(s)⊗CDX(resp. D(s) := C(s)⊗CD), with the filtration C(s)⊗CF, that we will also denote by F. Let p∈Dand let us write O=OX,p and D=DX,p. Let us consider the following properties (GDL)pFor some (or any)8reduced local equation f∈Oof Dat p, the ideal annD(s)fsis generated by order one operators, i.e. (see (8)) annD(s)fs= D(s)·Θf,s. (GDL) Property (GDL)pholds for any p∈D. It is clear that the property (GDL) is an open condition. (2.3.2) Remark. Since annD[s]fsis an ideal finitely generated of D[s], property (GDL)pholds if and only if there is a non zero polynomial β(s)∈C[s] such that β(s) annD[s]fs⊂ann(1) D[s]fs. A sufficient condition for the existence of a such polynomial is that the quotient annD[s]fs/ann(1) D[s]fsis holonomic as D-module. However this condition is not necessary as shown in example 3.1. It is clear that any divisor of differential linear type satisfies (GDL). (2.3.3) Example. The divisors of the examples in 3 satisfy (GDL) but are not of differential linear type. Let us assume that Xis an open subset of Cnwith coordinates x1,...,xn and that Dhas reduced global equation f:X→C. Morphism ϕin (7) is then given explicitly by ϕ:F(s, ξ)∈OX[s, ξ1,...,ξn]7→ F(f, f′ x)td∈ R(OX(f, f′ x1,...,f′ xn)) for each homogeneous polynomial Fof degree d, where ξidenotes the symbol of ∂ ∂xi. Let us denote by ker(1) ϕthe ideal of OX[s, ξ] generated by [ker ϕ]1≡Θf,s. To say that Dis of commutative linear type means that ker ϕ= ker(1) ϕ. (2.3.4) By shrinking Xif needed, let us take a system of generators of Der(log D) on X,δi=Pn j=1 aij ∂ ∂xj,1≤i≤m, with δi(f) = αif. In other words, (−αi, ai1,...,ain), 1 ≤i≤m, is a system of generators of the syzygies of f, f′ x1,...,f′ xn. Clearly the homogeneous polynomials ∆i=−αis+ai1ξ1+···+ ainξn, 1 ≤i≤m, generate ker(1) ϕ. In examples in Section 3 the divisors are not of differential linear type but they do satisfy the following property: 8One easily sees that this condition does not depend on the choice of the local equation. 19 (GCL) There is an integer N≥0 such that sNker ϕ⊂ker(1) ϕ. In other words, if we consider the map ϕ′:OX[s, s−1, ξ]−→ R(OX(f, f′ x))ft ⊂OX[t]ft induced by ϕ, property (GCL) exactly means that ker ϕ′is generated by ker(1) ϕ. Let us note that property (GCL) does not depend on the particular equation fand so it is a property of D. It is clear that any divisor of commutative linear type satisfies (GCL). (2.3.5) Proposition. If Dsatisfies property (GCL), then it is Euler homogeneous. Proof. The proof is analogous to the proof of proposition (2.2.5). Let N > 0 be such that sNker ϕ⊂ker(1) ϕ. We can proceed locally at each point p∈D. We know that fbelongs to the integral closure of the gradient ideal I= (f′ x1,...,f′ xn)⊂O(cf. [35], §0.5, 1)), i.e. there is an integer d > 0 and elements ai∈Id−isuch that fd+ad−1fd−1+···+a0= 0. Equivalently, there is a homogeneous polynomial F∈O[s, ξ] of degree d > 0 such that F(f, f′ x1,...,f′ xn) = 0 and F(s, 0,...,0) = sd. Then F∈ker ϕpand sNF=Pm i=1 Qi∆i, where the ∆ihave been defined in (2.3.4). By taking ξ1=··· =ξn= 0 we deduce that sN+d=−PiQi(s, 0. . . , 0)αisand so at least one of the αiis a unit, i.e. f∈(f′ x1,...,f′ xn) and Dis Euler homogeneous at p. (2.3.6) Corollary. If dim X= 2 and Dsatisfies (GCL), then it is of commutative linear type. (2.3.7) Remark. Proposition (2.2.13) suggests that property (GCL) could imply property (GK), but the divisor in example 3.2 satisfies (GCL) and not (GK). Proposition (2.2.9) suggests the following question. (2.3.8) Question. We do not know whether the implication (GCL) ⇒(GDL) is true or not. In the same vein, we do not know whether there is some relationship between Nand the degree of β(s) or not, where sNker ϕ⊂ker(1) ϕand β(s) annDX[s]fs⊂ann(1) DX[s]fs(see examples in 3) (2.3.9) Question. Concerning property (GDL), we do not know any “direct” algorithm9to find generators of the annihilator of fsover the Weyl algebra Q(s)[x, ∂], for a given polynomial f∈Q[x]. (2.3.10) Question. All the examples of free divisors satisfying the LCT we have been able to compute (see for instance examples in section 3) satisfy properties (GCL) and (GDL), but we do not know whether this a general fact or not. 9Not based on the known algorithms for computing the annihilator of fsover Q[s, x, ∂]. 20 (2.3.11) Question. Let b(s) be the Bernstein polynomial of fat a point10 p∈D. We have a functional equation b(s)fs=Pfs+1 with P∈D[s], which means that b(s)−P f ∈annD[s]fs. If Dsatisfies (GDL)pat p, then there is a non zero polynomial β(s)∈C[s] such that β(s) annD[s]fs⊂ann(1) D[s]fs, and so there is a functional equation b′(s)fs=P′fs+1 with b′(s)−P′f∈ ann(1) D[s]fs. In the examples in 3, property (GDL) is always satisfied (an also the LCT), but more precise properties hold. Namely, b(s) is a multiple of β(s) and the functional equation b(s)fs=Pfs+1 for the Bernstein polynomial already satisfies b(s)−Pf ∈ann(1) D[s]fswithout multiplying by β(s). We do not know whether these properties are a consequence of the property (GCL), and eventually of the LCT. On the other hand, a more precise relationship between the factors of β(s) and the tower ann(1) D[s]fs ann(2) D[s]fs ··· ann(k0) D[s]fs= annD[s]fs appears in examples in section 3.3. (2.3.12) Question. A natural question is to check whether linear free divisors [17], or locally weakly quasi-homogeneous free divisors (LWQH) [9] satisfy or not properties (GCL) or (GDL). For instance, all the divisors in section 3 are free LWQH. 3 Examples In this section we study some examples of free divisors in the affine space X= Cn, for n= 3,4 with a polynomial equation, from the point of view of properties (GCL), (GDL) and relatives. As a matter of notation, every time we have a global basis {δ1,...,δn}of the logarithmic derivations, we write σi=σ(δi)∈C[x][ξ]⊂Γ(X, Gr DX) = OX(X)[ξ]. The computations are done at the global level of Weyl algebras W=C[x, ∂] or W[s] or polynomial rings, and we have used [21] and [18]. 3.1 The example of “four lines” revisited In this example X=C3and D={f= 0}with f=x1x2(x1+x2)(x1+x2x3) (see [3, Remark 4.2.4], [5, ex. 6.2] and [4, 4]). There is a global basis of Der(log D), {δ1, δ2, δ3}, with δ1(f) = δ2(f) = 0, δ3(f) = 4f. Since (σ1, σ2) : σ3= (σ1, σ2, τ), with τa certain homogeneous polynomial in ξof degree 2, and (σ1, σ2) : τ= (x1, x2), we deduce that Dis not Koszul at any point of the line x1=x2= 0, which is strictly contained in the singular locus of D. 10A similar question can be considered in the global polynomial case. 21 The operators δ1, δ2, δ3−4sform a global OX-basis of F1 TannDX[s](fs) and their symbols with respect to the total order filtration σ1, σ2, σ3−4sform a regular sequence in C[x][ξ, s], and so, by flatness, in GrFTDX,p[s] = OX,p[ξ1, ξ2, ξ3, s] for any p∈X. Then, the divisor Dsatisfies property (GK). The kernel of ϕ:OX[s, ξ1, ξ2, ξ3]→ R(Jac(D)) is generated by σ1, σ2, τ, σ3− 4s. Moreover (ker ϕ)∩OX[ξ1, ξ2, ξ3] = OX[ξ](σ1, σ2, τ), τ /∈OX[ξ, s](σ1, σ2, σ3− 4s) and ker ϕ=ker(1) ϕ:f. We have sτ ∈(σ1, σ2, σ3−4s) and so s(ker ϕ)⊂ker(1) ϕ. In particular, D satisfies property (GCL). On the other hand, it is possible to lift τto an operator T∈Wsuch that σ(T) = τand T(fs) = 0. Then, we deduce that annDX[s]fs=DX[s](δ1, δ2, T, δ3−4s),annDXfs=DX(δ1, δ2, T ), and GrFTannDX[s]fs= ker ϕ, GrF(annDXfs) = (ker ϕ)∩OX[ξ] (i.e. Dsatisfies property (10)) and Dis not of differential linear type. We also have T f ∈W(δ1, δ2) and so annDXfs={A|Af ∈DX(δ1, δ2)}. We find that (s+ 1/2)T∈W[s](δ1, δ2, δ3−4s) and so (s+ 1/2) annDX[s]fs⊂ann(1) DX[s]fs,(11) i.e. Dsatisfies property (GDL). However, the DX-module M= annDX[s](fs)/ann(1) DX[s](fs) is not holonomic: we have M=DX·Pand a syzygy computation shows that annDX[s]P=DX[s] (x1, x2, s + 1/2) ,annDXP=DX(x1, x2) and so Mis not holonomic. The global Bernstein polynomial of fis bf(s) = (s+ 1)3(s+ 1/2)(s+ 3/4)(s+ 5/4). From (11) and the fact that the bf(s) does not have any integer root less than −1 we deduce that annDXf−1is generated by operators of order 1, even if D is not of differential linear type. Let us explain now “why” Dis pre-Spencer without testing the definition (1.5.2) itself. Let pbe any point of Dand let us write O=OX,p,D=DX,p. For the exactness of DX⊗V0Sp OXin degrees 6= 0 we can apply criterion (2.2.18). We know that σ1, σ2, σ3−4sis a regular sequence in GrFTD[s]. We 22 need to prove that D[s]/ann(1) D[s](fs) has no s-torsion. For that, let Q∈D[s] and suppose that sQ ∈ann(1) D[s](fs). It is clear that Q∈annD[s](fs) and we know from (11) that (s+ 1/2)Q∈ann(1) D[s](fs), but sQ, (s+ 1/2)Q∈ann(1) D[s](fs)⇒Q∈ann(1) D[s](fs). On the other hand, by specializing (11) at s= 0 we deduce that T∈D(δ1, δ2, δ3) and so (σ3) + (ker Φp)∩O[ξ] = (σ1, σ2, σ3, τ)⊂GrFD(δ1, δ2, δ3). But dim O[ξ] (σ1, σ2, σ3, τ)= dim O[ξ] (σ1, σ2, τ)−1 = dim R(Jac(D)p)−1 = 3 and D⊗D(log D)O=D/D(δ1, δ2, δ3) is holonomic. Finally, we check that there exists an operator Pin the Weyl algebra Wof order 6 such that the Bernstein functional equation bf(s)fs=Pfs+1 holds, but for which bf(s)−Pf is not on only in annW[s]fs, but in W[s](δ1, δ2, δ3−4s). 3.2 The example of “five lines” In this example X=C4and D={f= 0}with f=x1x2(x1+x2)(x1+ x2x3)(x1+x2x4). There is a global basis of Der(log D), {δ1, δ2, δ3, δ4}, with δ1(f) = δ2(f) = δ3(f) = 0, δ4(f) = 5f. We have a:= (σ1, σ2) : σ36= (σ1, σ2),(σ1, σ2) : a= (x1, x2) and so Dis neither Koszul nor satisfies property (GK)p(i.e. σ1, σ2, σ3, σ4−5s is not a regular sequence in GrFTDX,p[s] = OX,p[ξ, s]) at any pin the plane x1=x2= 0, which is strictly contained in the singular locus of D. So we cannot use criterion (2.2.18) to prove that Dis pre-Spencer. Nevertheless, a global computation in Wshows that the complex DX⊗V0Sp OXis exact in degrees 6= 0. The vanishing of h−1(DX⊗V0Sp OX) exactly means that the syzygies of δ1, δ2, δ3, δ4are generated by those of “Spencer type”, i.e. those coming from the relations expressing the brackets [δi, δj], 1 ≤i < j ≤4, as linear combinations of the δk, 1 ≤k≤4 (see the proof of proposition (2.2.17)). However, the syzygies of δ1, δ2, δ3are not generated by those of “Spencer type”. That represents a big difference with regular sequences in the commutative case. The kernel of ϕ:OX[s, ξ]→ R(Jac(D)) is generated by σ1, σ2, σ3, τ1, τ2, τ3, σ4− 5s, where the τi∈C[x][ξ] are homogeneous in ξof degree 2 and τi/∈ker(1) ϕ. 23 Moreover (ker ϕ)∩OX[ξ] = OX[ξ](σ1, σ2, σ3, τ1, τ2, τ3) and ker ϕ=ker(1) ϕ: f. As in example 3.1, we have s(ker ϕ)⊂ker(1) ϕand so Dsatisfies property (GCL). On the other hand, there are operators Ti∈annDXfssuch that σ(Ti) = τi, i= 1,2,3. We deduce that annDX[s]fs=DX[s](δ1, δ2, δ3, T1, T2, T3, δ4−5s), annDXfs=DX(δ1, δ2, δ3, T1, T2, T3) and GrFTannDX[s]fs= ker ϕ, GrF(annDXfs) = (ker ϕ)∩OX[ξ] (i.e. Dsatisfies property (10)) and Dis not of differential linear type. We also have Tif∈W(δ1, δ2, δ3) and so annDXfs={A|Af ∈DX(δ1, δ2, δ3)}. We also have the following (s+ 2/5) annD[s]fs⊂ann(1) D[s]fs,(12) i.e. Dsatisfies property (GDL). The global Bernstein polynomial of fis bf(s) = (s+ 1)4(s+ 2/5)(s+ 3/5)(s+ 4/5)(s+ 6/5). From (12) and the fact that the bf(s) does not have any integer root less than −1 we deduce that annDXf−1is generated by operators of order 1, even if D is not of differential linear type. To conclude that Dis Spencer, we have to prove that DX⊗V0OX= DX/DX(δ1, δ2, δ3, δ4) is holonomic, but this can be done in a completely similar way as in the example 3.1. Finally Dsatisfies the LCT after remark (1.7.3). 3.3 The family Dk={fk= (x1x3+x2)(xk 1−xk 2) = 0} This family of examples has been studied in [12]. All of them satisfy the LCT. Here we study their behavior with respect to properties (GCL), (GDL) and relatives. 3.3.1 The case k= 4 There is a global basis {δ1, δ2, δ3}of Der(log D4) with δ1(f4) = δ2(f4) = 0 and δ3(f4) = 5f4. We see that σ1, σ2is a regular sequence and so D4satisfies property (GK) (see proposition (2.2.16)) but a:= (σ1, σ2) : σ36= (σ1, σ2),(σ1, σ2) : a= (x1, x2) 24 and so D4is not Koszul at any pin the line x1=x2= 0, which is strictly contained in the singular locus of D4. The kernel of ϕ:OX[s, ξ]→ R(Jac D4) is generated by σ1, σ2, σ3−5s, τ where τis homogeneous in ξof degree 2 and τ /∈ker(1) ϕ. Moreover (ker ϕ)∩OX[ξ] = (σ1, σ2, τ) and ker ϕ=ker(1) ϕ:f4. In this example we find that s2(ker ϕ)⊂ker(1) ϕ(but s(ker ϕ)6⊂ ker(1) ϕ) and so D4satisfies property (GCL). In this example, property (10) does not hold, i.e. GrFTannDX[s]fs 46= ker ϕ, or equivalently GrFannDXfs 46= (ker ϕ)∩O[ξ]. Let us call Gr(l)annDXfs 4the ideal of Gr DXgenerated by GriannDXfs 4for i≤l, and ann(l) DXfs 4the left ideal of DXgenerated by FlannDXfs 4. We find the following facts: l= 2: Gr(2) annDXfs 4= (σ1, σ2, x1τ) and ann(2) DXfs 4=DX(δ1, δ2, T1) with σ(T1) = x1τ. We also find that the commutative relation sx1τ∈ker(1) ϕlifts to the differential relation (s+ 3/5)T1∈ann(1) DX[s]fs 4. Moreover, ann(1) DX[s]fs 4: (s+ 3/5) = DX[s](δ1, δ2, δ3−5s, T1) = ann(2) DX[s]fs 4. l= 3: Gr(3) annDXfs 4= (σ1, σ2, x1τ, ξ3τ) and ann(3) DXfs 4=DX(δ1, δ2, T1, T2) with σ(T2) = ξ3τ. We have (s+ 2/5)T2∈ann(2) DX[s]fs 4. Moreover, ann(2) DX[s]fs 4: (s+ 2/5) = ann(3) DX[s]fs 4. We check that annDX[s]fs 4= ann(3) DX[s]fs 4, annDXfs 4={A|Af4∈DX(δ1, δ2)} and δ1, δ2, T2is a (non involutive) basis of annDXfs 4. We have then that β(s) annDX[s]fs 4⊂ann(1) DX[s]fs 4, with β(s) = (s+3/5)(s+ 2/5), and D4satisfies property (GDL). We conclude as in example 3.1 that D4is Spencer. The global Bernstein polynomial of f4is bf4(s) = (s+ 1)3(s+ 6/5)(s+ 3/5)(s+ 4/5)(s+ 2/5) and it is a multiple of β(s). Although the elements f, δ1, δ2, δ3−5sare not an involutive basis of the ideal of DX[s] that they generate, it is possible to check, starting from the commutative relation s7∈C[x][s, ξ](f, σ1, σ2, σ3−5s), 25