The module Dfs for locally quasi-homogeneous free divisors

arXiv:math/0206262v1 [math.AG] 25 Jun 2002 The module Dfsfor locally quasi-homogeneous free divisors Francisco Calder´on-Moreno and Luis Narv´aez-Macarro∗ Abstract We find explicit free resolutions for the D-modules Dfsand D[s]fs/D[s]fs+1, where fis a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper. Introduction In this paper we study the module Dfs, where Dis the ring of germs at 0∈Cnof linear holomorphic differential operators and fis a reduced local equation of a locally quasi-homogeneous free divisor D⊂(Cn,0). The module Dfsencodes an enormous amount of geometric information of the singularity f= 0, but usually it is hard to work with in an explicit way. We prove the following results (see corollary 5.8 and theorem 5.9): (A) Let f= 0 be a reduced local equation of a locally quasi-homogeneous free divisor of Cn, and let {δ1,...,δn−1}be a basis of the module of vector fields vanishing on f. Then 1. The δigenerate the ideal AnnDfs. ∗The authors are supported by PB97-0723. 0Keywords: Free divisor, de Rham complex, D-module, locally quasi-homogeneous, Koszul complex, Spencer complex, ideal of linear type. 0Mathematical Subject Classification: 14F40, 32S20, 32S40. 1 2. There exist explicit Koszul-Spencer type free resolutions for the modules Dfsand D[s]fs/D[s]fs+1 built on δ1,...,δn−1and f, δ1,...,δn−1, respectively. Locally quasi-homogeneous free divisors form an important class of divisors with non isolated singularities: normal crossing divisors, the union of reflecting hyperplanes of a complex reflection group, free hyperplane arrangements or the discriminant of stable mappings in Mather’s “nice dimensions” are examples of such divisors. Let Xbe a complex analytic manifold. Given a divisor D⊂X, let us write j:U=X\D ֒→Xfor the corresponding open inclusion and Ω•(∗D) for the meromorphic de Rham complex with poles along D. In [11], Grothendieck proved that the canonical morphism Ω•(∗D)→Rj∗(CU) is an isomorphism (in the derived category). This result is usually known as (a version of) Grothendieck’s Comparison Theorem. In [17], K. Saito introduced the logarithmic de Rham complex associated with D, Ω• X(log D), generalizing the well known case of normal crossing divisors (cf. [8]). In the same paper, K. Saito also introduced the important notion of free divisor. In [7], it is proved that the logarithmic de Rham complex Ω• X(log D) computes the cohomology of the complement Uif Dis a locally quasihomogeneous free divisor (we say that Dsatisfies the logarithmic comparison theorem). In other words, the canonical morphism Ω• X(log D)→Rj∗(CU) is an isomorphism, or using Grothendieck’s result, the inclusion Ω• X(log D)֒→ Ω•(∗D) is a quasi-isomorphism. In fact, in [5] it is proved that, in the case of dim X= 2, Dis locally quasi-homogeneous if and only if it satisfies the logarithmic comparison theorem. Since the derived direct image Rj∗(CU) is a perverse sheaf (it is the de Rham complex of the holonomic module of meromorphic functions with poles along D[15], II, th. 2.2.4), we deduce that Ω• X(log D) is perverse for every locally quasi-homogeneous free divisor. On the other hand, the first author proved the following results [4]: Let D⊂Xbe a Koszul free divisor (see definition 1.6) and Ithe left ideal of the ring DXof differential operators on Xgenerated by the logarithmic vector fields with respect to D. Then 1) The left DX-module DX/Iis holonomic. 2 2) There is a canonical isomorphism in the derived category Ω• X(log D)≃RHomDX(DX/I,OX). As a consequence, the logarithmic de Rham complex associated with a Koszul free divisor is a perverse sheaf. The proof of (A) depends strongly on the following result, which has been suggested by the above results (see theorem 4.3): (B) Every locally quasi-homogeneous free divisor is Koszul free. In the first three sections we introduce some material concerning locally quasi-homogeneous free divisors, Koszul free divisors, the notion of linear type ideal and the module Dfs. In the fourth section we include the proof of (B) in our previous paper [6]. The fifth section is the main part of this paper and contains the proof of (A) and some related results. In the sixth section we study some examples and we state some problems and conjectures. The first part of (A) has been proposed (without proof) in [1, page 240] in the particular case of discriminants of versal deformations of simple hypersurface singularities. The normal crossing divisors case has been treated in [10]. 1 Locally quasi-homogeneous and Koszul free divisors 1.1 Let Xbe a n-dimensional complex analytic manifold. We denote by π:T∗X→Xthe cotangent bundle, OXthe sheaf of holomorphic functions on X,DXthe sheaf of linear differential operators on X(with holomorphic coefficients), GrF•(DX) the graded ring associated with the filtration F•by the order, σ(P) the principal symbol of a differential operator Pand {−,−} the Poisson bracket on OT∗Xor GrF•(DX). We will note O=OX,p,D=DX,p and GrF•(D) = GrF•(DX)pthe respective stalks at p, with pa point of X. If J⊂Dis a left ideal, we denote by σ(J) the corresponding graded ideal of GrF•(D). Given a divisor D⊂X, we denote by Der(log D) the OX- module of the logarithmic vector fields with respect to D[17]. If fis a local 3 equation of Dat p, we denote by Der(log f) the stalk at pof Der(log D), whose elements are germs at pof vector fields δsuch that δ(f)∈(f). Definition 1.2.– A divisor Dis Euler-homogeneous at p∈Dif there is a local equation hfor Daround p, and a germ of (logarithmic) vector field δ such that δ(f) = f. A such δis called a local Euler vector field for f. The set of points where a divisor is Euler-homogeneous is open. Definition 1.3.– (cf. [7]) A germ of divisor (D, p)⊂(X, p) is quasihomogeneous if there are local coordinates x1,...,xn∈OX,p with respect to which (D, p) has a weighted homogeneous defining equation (with strictly positive weights). A divisor Din a n-dimensional complex manifold Xis locally quasi-homogeneous if the germ (D, p) is quasi-homogeneous for each point p∈D. A germ of divisor (D, p)⊂(X, p) is locally quasi-homogeneous if the divisor Dis locally quasi-homogeneous in a neighborhood of p. Obviously a locally quasi-homogeneous divisor is Euler-homogeneous at every point. Definition 1.4.– We say that a reduced germ f∈OX,p is locally quasihomogeneous if the germ of divisor ({f= 0}, p) is. Remark 1.5.– A reduced germ f∈OX,p is locally quasi-homogeneous if and only if for every q∈ {f= 0}near pthere are local coordinates z1,...,zn∈OX,q and a quasi-homogeneous polynomial P(t1,...,tn) (with strictly positive weights) such that fq=P(z1,...,zn). Definition 1.6.– ([17], [4], def. 4.1.1) Let D⊂Xbe a divisor. We say that Dis free at p∈Xif Der(log D)pis a free O-module (of rank n). We say that Dis a Koszul free divisor at p∈Xif it is free at pand there exists a basis {δ1, . . . , δn}of Der(log D)psuch that the sequence of symbols {σ(δ1),...,σ(δn)}is regular in GrF•(D) = GrF•(DX)p. If Dis a free (resp. Koszul free) divisor at each point of X, we simply say that it is free (resp. Koszul free). We say that a reduced germ f∈OX,p is free if the divisor f−1(0) is free at p. Let’s remark that a divisor Dis automatically Koszul free at every p∈ X\D. Remark 1.7.– The ideal ID,p = GrF•(D)Der(log D)pis generated by the elements of any basis of Der(log D)p. As Dis Koszul free at pif and only if depth(ID,p,GrF•(D)) = n(cf. [14], cor. 16.8), it is clear that the definition 4 of Koszul free divisor does not depend on the election of a particular basis. By the coherence of GrF•(DX), if a divisor is Koszul free at a point, then it is Koszul free near that point. We have not found a reference for the following well known proposition (see [14], th. 17.4 for the local case). Proposition 1.8.– Let C{x}be the ring of convergent power series in the variables x=x1, . . ., xnand let Gbe the graded ring of polynomials in the variables ξ1, . . . , ξtwith coefficients in C{x}. A sequence σ1,...,σsof homogeneous polynomials in Gis regular if and only if the set of zeros V(I) of the ideal Igenerated by σ1,...,σshas dimension n+t−sin U×Ct, for some open neighborhood Uof 0 (then each irreducible component has dimension n+t−s). Proof: Let C{x, ξ}be the ring of convergent power series in the variables x1,...,xn, ξ1,...,ξt. As the σiare homogeneous in Gand the ring C{x, ξ} is a flat extension of G, the σiare a regular sequence in Gif and only if they are a regular sequence in C{x, ξ}. But the last condition is equivalent to the equality (loc. cit.): dim(0,0)(V(I)) = dim (C{x, ξ}/I) = n+t−s. Finally, using the fact that all the σiare homogeneous in the variables ξ, the local dimension of V(I) at (0,0) coincides with its dimension in U×Ctfor some neighborhood Uof 0. C.Q.D. Corollary 1.9.– Let D⊂Xbe a free divisor. Let Jbe the ideal in OT∗X generated by π−1Der(log D). Then, Dis Koszul free if and only if the set V(J) of zeros of Jhas dimension n(in this case, each irreducible component of V(J) has dimension n). Proposition 1.10.– Let Xbe a complex manifold of dimension nand let D⊂Xbe a divisor. Then: 1. Let X′=X×Cand D′=D×C. The divisor D⊂Xis Koszul free if and only if D′⊂X′is Koszul free. 2. Let Ybe another complex manifold of dimension rand let E⊂Ybe a divisor. Then: a) The divisor (D×Y)∪(X×E) is free if D⊂X and E⊂Yare free. 5 b) The divisor (D×Y)∪(X×E) is Koszul free if D⊂Xand E⊂Y are Koszul free. Proof: 1. It is a consequence of [7], lemma 2.2, (iv) and the fact that σ1,...,σn is a regular sequence in OX,p[ξ1,...,ξn] if and only if ξn+1, σ1,...,σnis a regular sequence in OX′,(p,t)[ξ1,...,ξn, ξn+1]. 2. a) It is an immediate consequence of Saito’s criterion (cf. [7], lemma 2.2, (v)). b) It is a consequence of a) and Corollary 1.9. C.Q.D. Example 1.11.– Examples of Koszul free divisors are: 1) Nonsingular divisors. 2) Normal crossing divisors. 3) Plane curves: If dimCX= 2, we know that every divisor D⊂Xis free [17], cor. 1.7. Let {δ1, δ2}be a basis of Der(log D)x. Their symbols {σ1, σ2} are obviously linearly independent over O, and by Saito’s criterion [17], 1.8, they are relatively primes in GrF•(D) = O[ξ1, ξ2]. So they form a regular sequence in GrF•(D), and Dis Koszul free (see [4], cor. 4.2.2). 4) Proposition 1.10 gives a way to obtain Koszul free divisors in any dimension. 5) There are irreducible Koszul free divisors in dimensions greater than 2, which are not constructed from divisors in lower dimension [16]: X=C3 and D≡ {f= 0}, with f= 28z3−27x2z2+ 24x4z+ 2432xy2z−22x3y2−33y4. A basis of Der(log f) is {δ1, δ2, δ3}, with δ1= 6y ∂x+ (8z−2x2)∂y−xy ∂z, δ2= (4x2−48z)∂x+ 12xy ∂y+ (9y2−16xz)∂z, δ3= 2x ∂x+ 3y ∂y+ 4z ∂z, and the sequence {σ(δ1), σ(δ2), σ(δ3)}is GrF•(D)-regular. 6 2 Ideals of linear type Definition 2.1.– (cf. [18], §7.2) Let Abe a commutative ring, I⊂A an ideal, R(I) = ⊕∞ i=0Idtd⊂A[t] the Rees algebra of Iand Sim(I) the symmetric algebra of the A-module I. We say that Iis of linear type if the canonical (surjective) morphism of graded A-algebras Sim(I)→ R(I) is an isomorphism. Lemma 2.2.– Given a commutative ring Aand an ideal I⊂Agenerated by a family of elements {ai}i∈Λ, the following properties are equivalent: a) Iis of linear type. b) If ϕ:A[{Xi}i∈Λ]→ R(I) is the morphism of graded algebras defined by ϕ(Xi) = ait, then the kernel of ϕis generated by homogeneous elements of degree 1. Proof: We consider the kernel of the surjective morphism of graded A- algebras Φ : A[{Xi}i∈Λ]→Sim(I),defined by Φ(Xi1···Xid) = ai1·...·aid. Then ker(Φ) = ker(ϕ) if and only if Iis of linear type, ker(Φ) is an ideal generated by its homogeneous elements of degree 1, ker(Φ)1, and ker(Φ)1= ker(ϕ)1. C.Q.D. The definition and the lemma above sheafify in the obvious way. The following results concern the case where the ideal Iis generated by a regular sequence. Lemma 2.3.– Let {a1,···, am}be an A-sequence. For p≤m, if αas1 1···asm m∈ (as1+k1 1,···, asp+kp p), then α∈(ak1 1,···, akp p). Proof: For j=p+ 1,···, m,{as1+k1 1,···, asp+kp p, asp+1 p+1 ,··· , asj j}is a regular A-sequence, and we can prove inductively that αas1 1···asj−1 j−1∈(as1+k1 1,···, asp+kp p). For i=p−1,··· ,0, {as1+k1 1,···, asi+ki i, aki+1 i+1 ,···, akp p}is a regular A-sequence, and we inductively prove that αas1 1···asi i∈(as1+k1 1,···, asi+ki i, aki+1 i+1 ,···, akp p). 7 C.Q.D. Proposition 2.4.– Let Abe a commutative ring and let I⊂Abe an ideal generated by a regular sequence a1,...,an. Then, the kernel of the morphism of graded algebras A[X1,...,Xn]→A[t], Xi7→ ait, is generated by aiXj−ajXi, 1 ≤i < j ≤n. In particular, Iis of linear type. Proof: Let gbe an homogeneous polynomial of degree min A[X1,···, Xn] such that g(a1,··· , an) = 0. Let expg=cXegbe the greatest monomial of gin the inverse lexicographic order, with eg= (s1,··· , st,0,···,0), st6= 0. Then g(X1,···, Xn)−expg∈(Xs1+1 1,···, Xst+1 t). By lemma 2.3, c=Pt−1 i=1 αiai∈(a1,···, at−1). Then f(X1,···, Xn) = g(X1,···, Xn)− t−1 X i=1 αi(aiXt−atXi)Xs1 1···Xsn n is an homogeneous polynomial of degree msuch that ef< egand f(X1,···, Xn)−g(X1,···, Xn)∈J= (aiXj−ajXi,0< i < j ≤n). In particular, f(a1,···, an) = 0. Consequently, after a finite number of steps, we will obtain h(X1) = cmXm 1, such that h(X1)−g(X1,···, Xn)∈J. So h(a1) = cmam 1= 0, cm= 0 and g(X1,··· , Xn)∈J. C.Q.D. 3 The module Dfs Let Xbe a n-dimensional complex analytic manifold, pa point in Xand f∈O=OX,p a non zero germ of holomophic function with f(p) = 0. Let Dbe the (germ of) divisor defined by f= 0. The free module of rank one over the ring O[f−1, s] generated by the symbol fshas a natural left module structure over the ring D[s] [2]: the action of a derivation δ∈DerC(O) is given by δ(fs) = δ(f)sf−1fs. 8 The following lemma is well-known and the proof is straightforward. Lemma 3.1.– For every linear differential operator P∈Dof order d, we have: P(fs) = CP,0fs+CP,1s 1fs−1+···+CP,ds dfs−d where CP,d =d!σ(P)(df) = {· · · {{σ(P), f}, f}d ···, f}. Denote by Jf⊂Othe jacobian ideal associated with f. The surjection δ∈DerC(O)7→ δ(f)∈Jf and the canonical isomorphism of graded O-algebras SimO(DerC(O)) ≃GrF•(D) (1) induce a surjective graded morphism of O-algebras ϕf: GrF•(D)−→ R(Jf).(2) In coordinates, GrF•(D) = O[ξ1,...,ξn], ξi=σ(∂i) and ϕf(σ(P)) = σ(P)(∂1(f)t,...,∂n(f)t) = σ(P)(df)td for every differential operator P∈Dof order d. The homogeneous part of degree 1 of ker ϕfis naturally identified with the O-module Θf={δ∈DerC(O)|δ(f) = 0} by means of the canonical isomorphism (1). Lemma 3.1 implies that σ(AnnDfs)⊂ker ϕf. Proposition 3.2.– With the above notations, if Jfis of linear type, then σ(AnnDfs) = ker ϕf and the left ideal AnnDfsof Dis generated by Θf. Proof: By lemma 2.2, ker ϕf= GrF•(D)Θf⊂σ(AnnDfs). The inclusion DΘf⊂AnnDfsis obvious. Let’s prove that AnnDfs⊂ DΘf. Clearly, F1AnnDfs= Θf. Suppose Fd−1AnnDfs⊂DΘfand take a 9 By induction hypothesis we know that the Koszul complex associated with ΞG′⊂GrF•DCn−1,0=On−1[ξ′ 1,...,ξ′ n−1] is exact or, equivalently, that σ′ 1,...,σ′ n−2, G′is a regular sequence in On−1[ξ′ 1,...,ξ′ n−1]. That implies that σ′ 1,...,σ′ n−2, ξ′ n, G =G′is a regular sequence in On[ξ′ 1,...,ξ′ n], i.e. that the Koszul complex associated with ΞG⊂GrF•DCn,0is exact, and the result is proved. For the second property, we filter the Spencer complex associated with Ξf⊂Das in [10], prop. 2.3.4: deg(Θf) = 1,deg(f) = 0. Its graded complex coincides with the Koszul complex associated with Ξf⊂ GrF•(D), and then the Spencer complex is exact. To conclude, we use corollary 5.8, (c). C.Q.D. 6 Examples and questions We know several (related) kind of free divisors: [LQH] Locally quasi-homogeneous (definition 1.3). [EH] Euler homogeneous (definition 1.2). [LCT] Free divisors satisfying the logarithmic comparison theorem. [KF] Koszul free (definition 1.6). [P] Free divisors such that the complex Ω• X(log D) is a perverse sheaf. We have then the following implications: [LQH] ⇒[EH] (obvious), [LQH] ⇒[LCT] by [7], th. 1.1, [LCT] ⇒[P], by [15], II, th. 2.2.4) [KF] ⇒[P] by [4], th. 4.2.1, [LQH] ⇒[KF] by theorem 4.3. Example 6.1.– (Free divisors in dimension 2) We recall theorem 3.9 from [5]: Let Xbe a complex analytic manifold of dimension 2 and D⊂Xa divisor. The following conditions are equivalent: 1. Dis Euler homogeneous. 16 2. Dis locally quasi-homogeneous. 3. The logarithmic comparison theorem holds for D. Consequently, in dimension 2 we have: [LQH] ⇔[EH] ⇔[LCT] and [KF] (cf. 1.11, 3) and [P] ([4]) always hold. In particular, [KF] 6⇒ [LQH], [EH], [LCT]. Examples of plane curves not satisfying logarithmic comparison theorem are, for instance, the curves of the family (cf. [5]): xq+yq+xyp−1= 0, p ≥q+ 1 ≥5. Example 6.2.– (An example in dimension 3) Let’s consider X=C3and D={f= 0}, with f=xy(x+y)(x+yz) [4]. A basis of Der(log D) is {δ1, δ2, δ3}, with δ1=xy∂x+y2∂y−4(x+yz)∂z, δ2=x(x+ 3y)∂x−y(3x+y)∂y+ 4x(z−1)∂z, δ3=x∂x+y∂y the determinant of the coefficients matrix being −16fand δ1(f) = 0, δ2(f) = 0, δ3(f) = 4f. In particular, Dis Euler homogeneous (E= (1/4)δ3) and we know [5] that it satisfies the logarithmic comparison theorem. Let I⊂OT∗Xbe the ideal generated by the symbols {σ1, σ2, σ3}of the basis of Der(log D). By corollary 1.9, Dis not Koszul free, because the dimension of V(I) at ((0,0, λ),0) ∈ T∗Xis 4, and neither is Dlocally quasi homogeneous. So: [LCT] 6⇒ [KF], [LQH], [EH] 6⇒ [KF], [LQH]. Finally, for the only missing relation, we quote the following conjecture from [5]: Conjecture 6.3.– If the logarithmic comparison theorem holds for D, then Dis Euler homogeneous. 17 Example 6.4.– Let’s see in the example 6.2 that the left ideal AnnD(fs) is not generated by Θfand then, Jfis not an ideal of linear type. Here, we set X=C3,p= (0,0,0) and E= (1/4)δ3. The O-modules Θfand Der(log f) = Θf⊕O·Eare generated by {δ1, δ2}and {δ1, δ2, E}, respectively. The symbols σ1=σ(δ1), σ2=σ(δ2) form a GrF•(D)-regular sequence (the proof is analogous to example 1.11, 3)). Then, as in the proof of [4], prop. 4.1.2, we have σ(DΘf) = GrF•(D)σ(Θf). For P= 2y2∂x∂y−2y2∂2 y−(2xz + 6yz)∂x∂z+ 10yz∂y∂z+ 8z(1 −z)∂2 z+ +(2x−4y)∂y∂z−x∂x−y∂y−8z∂z+ 4∂z and R=C[x, y, z], S=R[ξ1, ξ2, ξ3], m=R(x, y, z) we check that 1. P∈AnnDX(fs), 2. (S(σ1, σ2) : σ(P)) = S(x, y), and then (S(σ1, σ2) : σ(P))∩R=R(x, y). So, σ(P)/∈Rm[ξ1, ξ2, ξ3]σ(Θf) and, by faithful flatness, σ(P)/∈O[ξ1, ξ2, ξ3]σ(Θf) = GrF•(D)σ(Θf). We conclude that P /∈DΘf. Problem 6.5.– We do not know whether a free divisor defined by a quasihomogeneous polynomial (with strictly positive weights) is locally quasihomogeneous. Problem 6.6.– We do not know any example of a free divisor D⊂Xwhose logarithmic de Rham complex Ω• X(log D) is not perverse. References [1] A.G. Aleksandrov. Nonisolated hypersurface singularities. 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