Locally quasi-homogeneous free divisors are Koszul free

ТРУДЫ МАТЕМАТИЧЕСКОГО ИНСТИТУТА ИМ.В.А.СТЕКЛОВА, 2002, т. 238, с. 81–85 УДК 512.7+517.55 Locally Quasi-Homogeneous Free Divisors Are Koszul Free1 c 2002 г.F.Calder´on-Moreno2,L.Narv´aez-Macarro3 Поступило в ноябре 2000 г. Let Xbe a complex analytic manifold and D⊂Xbe a free divisor. If Dis locally quasi-homogeneous, then the logarithmic de Rham complex associated to Dis quasi-isomorphic to Rj∗(CX\D), which is a perverse sheaf. On the other hand, the logarithmic de Rham complex associated to a Koszul-free divisor is perverse. In this paper, we prove that every locally quasi-homogeneous free divisor is Koszul free. 1. INTRODUCTION Let Xbe a complex analytic manifold. For a divisor D⊂X,letuswritej:U=X\D→X for the corresponding open inclusion and Ω•(∗D) for the meromorphic de Rham complex with poles along D. In [5], Grothendieck proved that the canonical morphism Ω•(∗D)→Rj∗(CU)isan isomorphism (in the derived category). This result is usually known as (a version of) Grothendieck’s Comparison Theorem. In [9], K. Saito introduced the subcomplex Ω• X(log D)ofΩ •(∗D), which he called a logarithmic de Rham complex associated to D, generalizing the well-known case of normal crossing divisors (see [4]). In the same paper, K. Saito also introduced the important notion of free divisor. In [3], it is proved that the logarithmic de Rham complex Ω• X(log D) computes the cohomology of the complement Uif Dis a locally quasi-homogeneous 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)isaquasiisomorphism. In fact, in [2] it is proved that, in the case of dim X=2,Dis locally quasihomogeneous 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[7, II, Theorem 2.2.4]), we deduce that the logarithmic comparison theorem for a free divisor Dimplies that the logarithmic de Rham complex associated to Dis a perverse sheaf. On the other hand, the first author proved in [1] the following results. Let D⊂Xbe a Koszulfree divisor (see Definition 2.3) and Ibe the left ideal of the ring DXof differential operators on X generated by the logarithmic vector fields with respect to D. Then, 1) The left DX-module DX/Iis holonomic. 2) There is a canonical isomorphism in the derived category Ω• X(log D)≃RHomDX(DX/I,OX). As a consequence of these results, the logarithmic de Rham complex associated to a Koszul-free divisor is a perverse sheaf. 1The research was partially supported by PB97-0723 and INTAS 97-1644. 2Departamento de ´ Algebra, Facultad de Matem´aticas, Universidad de Sevilla, Ap 1160, 41080 Sevilla, Spain. E-mail: [email protected] 3Departamento de ´ Algebra, Facultad de Matem´aticas, Universidad de Sevilla, Ap 1160, 41080 Sevilla, Spain. E-mail: narv[email protected]s.es 681 82 CALDER´ ON-MORENO, NARV´ AEZ-MACARRO In this paper, we prove the following result, suggested by the previous ones: every locally quasi-homogeneous free divisor is Koszul free (see Theorem 3.2). At the end, we study some examples in dimensions two and three. 2. PRELIMINARY RESULTS Let Xbe an n-dimensional complex analytic manifold. We denote by π:T∗X→Xthe cotangent bundle, by OXthe sheaf of holomorphic functions on X,byDXthe sheaf of linear differential operators on X(with holomorphic coefficients), by GrF•(DX) the graded ring associated to the filtration by the order, and by σ(P) the principal symbol of a differential operator P.We will denote by O=OX,x,D=DX,x,andGr F•(D)=GrF•(DX)xthe respective stalks at x,with apointxin X.LetD⊂Xbe a hypersurface. We denote by Der(log D)theOX-module of the logarithmic vector fields with respect to D[9]. Definition 2.1. A divisor Dis Euler homogeneous at xif there are a local equation hfor D around xand a germ of logarithmic vector field δsuch that δ(h)=h. The set of points where a divisor is Euler homogeneous is open. Definition 2.2 (see [3]).A divisor Din an n-dimensional complex manifold Xis locally quasihomogeneous if, at each point q∈D, there are local coordinates (U;x1,...,x n) centered at q(i.e., with xi(q)=0fori=1,...,n) with respect to which D∩Uhas a weighted homogeneous defining equation (with strictly positive weights). Obviously, a locally quasi-homogeneous divisor is Euler homogeneous at every point. Definition 2.3 [1, Definition 4.1.1].Let D⊂Xbe a divisor. We say that Dis a Koszul-free divisor at xif there exists a basis {δ1,...,δ n}of Der(log D)xsuch that the sequence of symbols {σ(δ1),...,σ(δn)}is regular in GrF•(D)=GrF•(DX)x.IfDis a Koszul-free divisor at each point of D, we simply say that it is a Koszul-free divisor. Remark 2.4. The ideal ID,x =Gr F•(D)Der(log D)xis generated by the elements of any basis of Der(log D)x.SinceDis Koszul free at xif and only if depth(ID,x,GrF•(D)) = n(see [6, Corollary 16.8]), it is clear that the definition of a Koszul-free divisor does not depend on the choice 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 this point. We have not found a reference for the following well-known proposition (see [6, Theorem 17.4] for the local case). Proposition 2.5. Let C{x}be the ring of convergent power series in the variables x= (x1,...,x n),and let Gbe the graded ring of polynomials in the variables ξ1,...,ξ twith coefficients in C{x}. A sequence σ1,...,σ sof homogeneous polynomials in Gis regular if and only if the set of zeros V(I)of the ideal Igenerated by σ1,...,σ shas dimension n+t−sin U×Ctfor 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,...,x n,ξ 1,...,ξ t. Since the σiare homogeneous and the ring C{x, ξ}is a flat extension of G,theσiare 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 [6, Theorem 17.4] dim(0,0)(V(I)) = dim(C{x, ξ}/I)=n+t−s. Finally, since all 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.  Corollary 2.6. Let D⊂Xbe a free divisor. Let Jbe the ideal in OT∗Xgenerated 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). ТРУДЫ МАТЕМАТИЧЕСКОГО ИНСТИТУТА ИМ.В.А.СТЕКЛОВА, 2002, т. 238 LOCALLY QUASI-HOMOGENEOUS FREE DIVISORS 83 Proposition 2.7. Let Xbe a complex manifold of dimension nand let D⊂Xbe a divisor. Then, the following assertions are valid. 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⊂Xand E⊂Yare free;and b) the divisor (D×Y)∪(X×E)is Koszul free if D⊂Xand E⊂Yare Koszul free. Proof. 1. It is a consequence of [3, Lemma 2.2(iv)] and the fact that σ1,...,σ nis a regular sequence in OX,p[ξ1,...,ξ n] if and only if ξn+1,σ 1,...,σ nis a regular sequence in OX,(p,t)[ξ1,...,ξ n,ξ n+1]. 2. a) It is an immediate consequence of Saito’s Criterion (see [3, Lemma 2.2(v)]). b) It is a consequence of a) and Corollary 2.6.  Example 2.8. Let us consider examples of Koszul-free divisors. 1) Nonsingular divisors. 2) Normal crossing divisors. 3) Plane curves: If dimCX= 2, we know that every divisor D⊂Xis free [9, Corollary 1.7]. Let {δ1,δ 2}be a basis of Der(log D)x.Theirsymbols{σ1,σ 2}are obviously linearly independent over O, and, by Saito’s Criterion [9, 1.8], they are relatively prime in GrF•(D)=O[ξ1,ξ 2]. So, they form a regular sequence in GrF•(D), and Dis Koszul free (see [1, Corollary 4.2.2]). 4) Proposition 2.7 gives a way to obtain Koszul-free divisors in any dimension. 5) There are irreducible Koszul-free divisors Yin dimensions greater than two, which are not normal crossing and do not have nontrivial factors [8]; for example, X=C3and Y≡{f=0}, with f=2 8z3−27x2z2+2 4x4z+2 432xy2z−22x3y2−33y4. A basis of Der(log f)is{δ1,δ 2,δ 3},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. 3. MAIN RESULTS Proposition 3.1. Let Dbe a free divisor in an analytic manifold Xand let Σ⊂Dbe a discrete set of points. If Dis Koszul free at every point x∈D\Σ,then Dis Koszul free (at every point of D). Proof. Let p∈Σandlet{δ1,...,δ n}be a basis of the logarithmic derivations of Dat p. By Corollary 2.6, we have to prove that the symbols σi=σ(δi) define an analytic set V= V(σ1,...,σ n)⊂π−1(U)ofdimensionn=dimXfor some open neighborhood U⊂Xof p.LetU be an open neighborhood of psuch that U∩Σ={p}. By hypothesis, we know that Dis Koszul free in U\{p}, and so (Corollary 2.6) the dimension of V∩π−1(U\{p})=V\T∗ pXis n.Now,let Wbe an irreducible component of V. It has, at least, dimension n.IfWis contained in T∗ pX,then it must be equal to T∗ pX, and dimW=n.Ifnot,dimW=dim(W\T∗ pX)≤dim(V\T∗ pX)=n. So, we conclude that Vhas dimension n. ТРУДЫ МАТЕМАТИЧЕСКОГО ИНСТИТУТА ИМ.В.А.СТЕКЛОВА, 2002, т. 238 6* 84 CALDER´ ON-MORENO, NARV´ AEZ-MACARRO Theorem 3.2. Every locally quasi-homogeneous free divisor is Koszul free. Proof. We proceed by induction on the dimension tof the ambient manifold X.Fort=1, the theorem is trivial, and, for t= 2, the theorem is directly proved in Example 2.8, 3). Now, we suppose that the result is true for t<n,andletDbe a locally quasi-homogeneous free divisor of a complex analytic manifold Xof dimension n.Letp∈D,andlet{δ1,...,δ n}be a basis of the logarithmic derivations of Dat p. Thanks to [3, Proposition 2.4 and Lemma 2.2(iv)], there is an open neighborhood Uof psuch that, for each q∈U∩Dwith q=p, the germ of pair (X, D,q) is isomorphic to a product (Cn−1×C,D×C,(0,0)), where Dis a locally quasi-homogeneous free divisor. The induction hypothesis implies that Dis a Koszul-free divisor at 0. Then, by assertion 1 of Proposition 2.7, Dis a Koszul-free divisor at qtoo. We have then proved that Dis a Koszul-free divisor in U\{p}. We conclude by using Proposition 3.1.  Corollary 3.3. Every free divisor that is locally quasi-homogeneous at the complement of a discrete set is Koszul free. In particular, the last corollary gives rise to a new proof of the fact that every divisor in dimension two is Koszul free (see Example 2.8, 3)). 4. EXAMPLES We know several (related) kinds of free divisors: [LQH] Locally quasi-homogeneous (Definition 2.2). [EH] Euler homogeneous (Definition 2.1). [LCT] Free divisors satisfying the logarithmic comparison theorem. [KF] Koszul free (Definition 2.3). [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 [3, Theorem 1.1], [LCT] ⇒[P] by [7, II, Theorem 2.2.4], [KF] ⇒[P] by [1, Theorem 4.2.1], [LQH] ⇒[KF] by Theorem 3.2. Example 4.1 (free divisors in dimension two).We recall Theorem 3.9 from [2]. Let Xbe a complex analytic manifold of dimension two and D⊂Xbe a divisor. The following conditions are equivalent: 1. Dis Euler homogeneous. 2. Dis locally quasi-homogeneous. 3. The logarithmic comparison theorem holds for D. Consequently, in dimension two we have [LQH] ⇔[EH] ⇔[LCT] and [KF] and [P] always hold (see Example 2.8, 3)). In particular, [KF] ⇒ [LQH], [EH], [LCT]. ТРУДЫ МАТЕМАТИЧЕСКОГО ИНСТИТУТА ИМ.В.А.СТЕКЛОВА, 2002, т. 238 LOCALLY QUASI-HOMOGENEOUS FREE DIVISORS 85 Examples of plane curves not satisfying logarithmic comparison theorem are, for instance, the curves of the family (see [2]) xq+yq+xyp−1=0,p≥q+1≥5. Example 4.2 (an example in dimension three).Let us consider X=C3and D={f=0}, with f=xy(x+y)(y+zx)[1].AbasisofDer(log D)is{δ1,δ 2,δ 3}with δ1=x∂x+y∂y, δ2=x2∂x−y2∂y−z(x+y)∂z, δ3=(xz +y)∂z, the determinant of the coefficients matrix being −fand δ1(f)=4f, δ2(f)=(2x−3y)f, δ3(f)=xf. In particular, Dis Euler homogeneous and satisfies the logarithmic comparison theorem [2]. Let I⊂O T∗Xbe the ideal generated by the symbols {σ1,σ 2,σ 3}of the basis of Der(log D). By Corollary 2.6, Dis not Koszul free, because the dimension of V(I)at((0,0,λ),0) ∈T∗X(λ=0) is greater than three. So, Dis not locally quasi-homogeneous either. Thus, [LCT] ⇒ [KF], [LQH],[EH] ⇒ [KF], [LQH]. 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