Algebraic computation of some intersection D-modules

arXiv:math/0604287v1 [math.AG] 12 Apr 2006 Algebraic computation of some intersection D-modules F. J. Calder´on Moreno and L. Narv´aez Macarro∗ March, 2006 Abstract Let Xbe a complex analytic manifold, D⊂Xa locally quasi-homogeneous free divisor, Ean integrable logarithmic connection with respect to Dand Lthe local system of the horizontal sections of Eon X− D. In this paper we give an algebraic description in terms of Eof the regular holonomic DX-module whose de Rham complex is the intersection complex associated with L. As an application, we perform some effective computations in the case of quasi-homogeneous plane curves. Introduction On a complex analytic manifold, intersection complexes associated with irreducible local systems on a dense open regular subset of a closed analytic subspace are the simple pieces which form any perverse sheaf. The RiemannHilbert correspondence allows us to consider the regular holonomic D-modules which correspond to these intersection complexes, that we call “intersection D-modules”. They are the simple pieces which form any regular holonomic D-module. Whereas intersection complexes are topological objects, intersection D-modules are algebraic: they are given by a system of partial linear differential equations with holomorphic coefficients. Intersection complexes can be constructed by an important operation: the intermediate direct image. Its description in terms of Verdier duality and usual derived direct images can be algebraically interpreted in the category of holonomic regular D-modules by using the deep properties of the de Rham functor. We need to compute localizations and D-duals. This can be effectively done, in principle, by using the general available algorithms in [25, 27, 26], but in the case of integrable logarithmic connections along a locally quasi-homogeneous free divisor, we exploit the logarithmic point of view [2, 4, 5, 8, 9, 30, 31] to previously obtain a general algebraic description of their associated intersection D-modules, from which we can easily derive effective computations. The main ingredients we use are the duality theorem proved in [5] and the logarithmic comparison theorem for arbitrary integrable logarithmic connections proved in [6], both with respect to locally quasi-homogeneous free divisors. ∗The authors are partially supported by MTM2004-07203-C02-01 and FEDER. 1 The algorithmic treatment of the computations in this paper will be developed elsewhere. Let us now comment on the content of this paper. In section 1 we remind the reader of the basic notions and notations and we review our previous results on logarithmic D-modules with respect to free divisors. We recall the logarithmic comparison theorem for arbitrary integrable logarithmic connections from [6], and we give the theorem describing the intersection D-module associated with an integrable logarithmic connection along a locally quasi-homogeneous free divisor. In section 2, given a locally quasi-homogeneous free divisor Dwith a reduced local equation f= 0 and a cyclic integrable logarithmic connection Ewith respect to D, we explicitly describe a presentation of D[s]·(Efs) over D[s] in terms of a presentation of Eover the ring of logarithmic differential operators. This description will be useful in order to compute the Bernstein-Sato polynomials associated with E. In section 3, the general results of the previous section are explicitly written down in the case of a family of integrable logarithmic connections with respect to a quasi-homogeneous plane curves. In section 4 we perform some explicit computations with respect to a cusp. We wish to thank H´el`ene Esnault who, because of a question about our paper [5], drew our attention to computing intersection D-modules. We also thank Tristan Torrelli for helpful information about the Bernstein-Sato functional equations and for some comments on a previous version of this paper. 1 Logarithmic connections with respect to a free divisor: theoretical set-up Let Xbe a n-dimensional complex analytic manifold and D⊂Xa hypersurface, and let us denote by j:U=X−D ֒→Xthe corresponding open inclusion. We say that Dis a free divisor [28] if the OX-module Der(log D) of logarithmic vector fields with respect to Dis locally free (of rank n), or equivalently if the OX-module Ω1 X(log D) of logarithmic 1-forms with respect to Dis 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), discriminant of stable mappings or bifurcation sets are examples of free divisors. We say that Dis quasi-homogeneous at p∈Dif there 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. We say that Dis locally quasi-homogeneous if it is so at each point p∈D. Let us denote by DX(log D) the 0-term of the Malgrange-Kashiwara filtration with respect to Don the sheaf DXof linear differential operators on X. When Dis a free divisor, the first author has proved in [2] that DX(log D) is the universal enveloping algebra of the Lie algebroid Der(log D), and then it is coherent and has noetherian stalks of finite global homological dimension. Locally, if {δ1,...,δn}is a local basis of the logarithmic vector fields on an open set V, any differential operator in Γ(V, DX(log D)) can be written in a unique 2 way as a finite sum X α∈Nn |α|≤d aαδα1 1···δαn n, where the aαare holomorphic functions on V. From now on, let us assume that Dis a free divisor. We say that Dis a Koszul free divisor [2] at a point p∈Dif the symbols of any (some) local basis {δ1,...,δn}of Der(log D)pform a regular sequence in Gr DX,p. We say that Dis a Koszul free divisor if it is so at any point p∈D. Actually, as M. Schulze pointed out, Koszul freeness is equivalent to holonomicity in the sense of [28]. Plane curves and locally quasi-homogeneous free divisors (e.g. free hyperplane arrangements or discriminant of stable mappings in Mather’s “nice dimensions”) are example of Koszul free divisors [3]. Alogarithmic connection with respect to Dis a locally free OX-module E endowed with: -) a C-linear morphism (connection) ∇′:E−→ E⊗OXΩ1 X(log D), satisfying ∇′(ae) = a∇′(e) + e⊗da, for any section aof OXand any section eof E, or equivalently, with -) a left OX-linear morphism ∇: Der(log D)−→ EndCX(E) satisfying the Leibniz rule ∇(δ)(ae) = a∇(δ)(e)+δ(a)e, for any logarithmic vector field δ, any section aof OXand any section eof E. The integrability of ∇′is equivalent to the fact that ∇preserve Lie brackets. Then, we know from [2] that giving an integrable logarithmic connection on a locally free OX-module Eis equivalent to extending its original OX-module structure to a left DX(log D)-module structure, and so integrable logarithmic connections are the same as left DX(log D)-modules which are locally free of finite rank over OX. Let us denote by OX(⋆D) the sheaf of meromorphic functions with poles along D. It is a holonomic left DX-module. The first examples of integrable logarithmic connections (ILC for short) are the invertible OX-modules OX(mD)⊂OX(⋆D), m∈Z, formed by the meromorphic functions hsuch that div(h) + mD ≥0. 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 we have the following local presentation over DX,p(log D) ([2], th. 2.1.4) OX,p(mD)≃DX,p(log D)/DX,p(log D)(δ1+mα1,...,δn+mαn).(1) (1.1) 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 DX(log D)-module, where the action of logarithmic vector fields is given by (δh)(e) = −h(δe) + δ(h(e)), δ (e⊗a) = (δe)⊗a+e⊗δ(a) (2) for any logarithmic vector field δ, any local section hof HomOX(E,OX), any local section eof Eand any local section aof OX(mD) (cf. [5], §2). Then E(mD) and E∗are ILC again, and the usual isomorphisms E(mD)(m′D)≃E((m+m′)D),E(mD)∗≃E∗(−mD) 3 are DX(log D)-linear. (1.2) If Dis Koszul free and Eis an ILC, then the complex DX L ⊗DX(log D)E is concentrated in degree 0 and its 0-cohomology DX⊗DX(log D)Eis a holonomic DX-module (see [5], prop. 1.2.3). If Eis an ILC, then E(⋆D) is a meromorphic connection (locally free of finite rank over OX(⋆D)) and then it is a holonomic DX-module (cf. [20], th. 4.1.3). Actually, E(⋆D) has regular singularities on the smooth part of D(it has logarithmic poles! [10]) and then it is regular everywhere [19], cor. 4.3-14, which means that if Lis the local system of horizontal sections of Eon U=X−D, the canonical morphism Ω• X(E(⋆D)) −→ Rj∗L is an isomorphism in the derived category. For any ILC E, or even for any left DX(log D)-module (without any finiteness property over OX), one can define its logarithmic de Rham complex Ω• X(log D)(E) in the classical way (cf. [10, def. I.2.15]), which is a subcomplex of Ω• X(E(⋆D)). It is clear that both complexes coincide on U. For any ILC Eand any integer m,E(mD) is a sub-DX(log D)-module of the regular holonomic DX-module E(⋆D), and then we have a canonical morphism in the derived category of left DX-modules ρE,m :DX L ⊗DX(log D)E(mD)→E(⋆D), given by ρE,m(P⊗e′) = Pe′. Since E(m′D)(mD) = E((m+m′)D) and E(m′D)(⋆D) = E(⋆D), we can identify morphisms ρE(m′D),m and ρE,m+m′. For any bounded complex Kof sheaves of C-vector spaces on X, let us denote by K∨=RHomCX(K,CX) its Verdier dual. The dual local system L∨appears as the local system of the horizontal sections of the dual ILC E∗. We have the following theorem (see [5, th. 4.1] and [6, th. (2.1.1)]): (1.3) Theorem. Let Ebe an ILC (with respect to the divisor D) and let L be the local system of its horizontal sections on U=X−D. The following properties are equivalent: 1) The canonical morphism Ω• X(log D)(E)→Rj∗Lis an isomorphism in the derived category of complexes of sheaves of complex vector spaces. 2) The inclusion Ω• X(log D)(E)֒→Ω• X(E(⋆D)) is a quasi-isomorphism. 3) The morphism ρE,1:DX L ⊗DX(log D)E(D)→E(⋆D)is an isomorphism in the derived category of left DX-modules. 4) The complex DX L ⊗DX(log D)E(D)is concentrated in degree 0and the DXmodule DX⊗DX(log D)E(D)is holonomic and isomorphic to its localization along D. Moreover, if Dis a Koszul free divisor, the preceding properties are also equivalent to: 4 5) The canonical morphism j!L∨→Ω• X(log D)(E∗(−D)) is an isomorphism in the derived category of complexes of sheaves of complex vector spaces. For Da locally quasi-homogeneous free divisor and E=OX, the equivalent properties in theorem (1.3) hold: this is the so called “logarithmic comparison theorem” [7] (see also [5, th. 4.4] and [6, cor. (2.1.3)] for other proofs based on D-module theory). (1.4) Let Ebe an ILC (with respect to D) and pa point in D. Let f∈ O=OX,p be a reduced local equation of Dand let us write D=DX,p,V0= DX(log D)pand E=Ep. We know from [6, lemma (3.2.1)] that the ideal of polynomials b(s)∈C[s] such that b(s)Efs⊂D[s]·Efs+1⊂E[f−1, s]fs is generated by a non constant polynomial bE,p(s). By the coherence of the involved objects we deduce that bE,q(s)|bE,p(s) for q∈Dclose to p. If bE,p(s) has some integer root, let us call κ(E, p) the minimum of those roots. If not, let us write κ(E, p) = +∞. Let us call κ(E) = inf{κ(E, p)|p∈D} ∈ Z∪ {±∞}. From now on let us suppose that Dis a locally quasi-homogeneous free divisor. (1.5) Theorem. Under the above hypothesis, if κ(E)>−∞, then the morphism ρE,k :DX L ⊗DX(log D)E(kD)−→ E(⋆D) (3) is an isomorphism in the derived category of left DX-modules, for all k≥ −κ(E). Proof. It is a straightforward consequence of [3], [4, th. 5.6] and theorem (3.2.6) of [6] and its proof. Q.E.D. Let us note that the hypothesis κ(E)>−∞ in theorem (1.5) holds locally on X. In the situation of theorem (1.5), if Lis the local system of the horizontal sections of Eon U=X−D, then the derived direct image Rj∗Lis canonically isomorphic (in the derived category) to the de Rham complex of the holonomic DX-module DX⊗DX(log D)E(kD): DR DX⊗DX(log D)E(kD)= DR DX L ⊗DX(log D)E(kD)≃ DR E(⋆D)≃Ω• X(E(⋆D)) ≃Rj∗L. Proceeding as above for the dual ILC E∗, we find that if κ(E∗)>−∞, then we have that the canonical morphism DR DX⊗DX(log D)E∗(k′D)−→ Rj∗L∨ is an isomorphism in the derived category for k′≥ −κ(E∗). 5 Let us denote by E,k,k′:DX⊗DX(log D)E((1 −k′)D)−→ DX⊗DX(log D)E(kD),(4) the DX-linear morphism induced by the inclusion E((1−k′)D)⊂E(kD), 1−k′≤ k, and by ICX(L) the intersection complex of Deligne-Goresky-MacPherson associated with L, which is described as the intermediate direct image j!∗L, i.e. the image of j!L→Rj∗Lin the category of perverse sheaves (cf. [1], def. 1.4.22). The following theorem describes the “intersection DX-module” corresponding to ICX(L) by the Riemann-Hilbert correspondence of Mebkhout-Kashiwara [13, 16, 17]. (1.6) Theorem. Under the above hypothesis, we have a canonical isomorphism in the category of perverse sheaves on X, ICX(L)≃DR (Im E,k,k′), for k≥ −κ(E),k′≥ −κ(E∗)and 1−k′≤k. Proof. Using our duality results in [5, §3], the Local Duality Theorem for holonomic DX-modules ([18], ch. I, th. (4.3.1); see also [22]) and theorem (1.5), we obtain DR DX⊗DX(log D)E((1 −k′)D)≃DR DX⊗DX(log D)E∗(k′D)∗(D)≃ DR DDXDX⊗DX(log D)E∗(k′D)≃DR DX⊗DX(log D)E∗(k′D)∨≃ [Rj∗L∨]∨≃j!L. On the other hand, the canonical morphism j!L→Rj∗Lcorresponds, through the de Rham functor, to the DX-linear morphism E,k,k′, and the theorem is a consequence of the Riemann-Hilbert correspondence which says that the de Rham functor establishes an equivalence of abelian categories between the category of regular holonomic DX-modules and the category of perverse sheaves on X. Q.E.D. (1.7) Remark. For E=OX, one has E∗=OXand there are examples where morphisms ρOX,k in (3) are never isomorphisms ([5], ex. 5.3). Nevertheless, for k=k′= 1 the image of the morphism OX,1,1:DX⊗DX(log D)OX−→ DX⊗DX(log D)OX(D) is always (canonically isomorphic to) OX, which is the regular holonomic DXmodule corresponding by the Riemann-Hilbert correspondence to ICX(CU) = CX, where CUis the local system of horizontal sections of OXon U. To see this, let us work locally as in (1). Then, morphism OX,1,1is given at point p by P∈DX,p/DX,p(δ1,...,δn)7→ Pf ∈DX,p/DX,p(δ1+α1,...,δn+αn) and the stalk at pof Im OX,1,1is given by DX,p/J where Jis the left ideal J={P∈DX,p |P f ∈DX,p(δ1+α1,...,δn+αn)}. 6 By Saito’s criterion [28] we can suppose    δ1 . . . δn   =A   ∂ ∂x1 . . . ∂ ∂xn    where Ais a n×nmatrix with entries in OX,p and det A=f. Writing B= adj(A)twe obtain B   δ1 . . . δn   =f   ∂ ∂x1 . . . ∂ ∂xn    eval. on f B   α1 . . . αn   =   ∂f ∂x1 . . . ∂f ∂xn   . Then    ∂ ∂x1 . . . ∂ ∂xn   f=f   ∂ ∂x1 . . . ∂ ∂xn   +   ∂f ∂x1 . . . ∂f ∂xn   =···=B   δ1+α1 . . . δn+αn    and ∂ ∂xi∈Jfor i= 1,...,n. Since Jis is not the total ideal, we deduce by maximality that Jis the ideal generated by the ∂ ∂xiand DX,p/J ≃OX,p. To conclude, one easily sees, from the fact that morphism OX,1,1factors through a∈OX7→ 1⊗a∈DX⊗DX(log D)OX(D) [it is DX-linear since, for any derivation δand any holomorphic function a,δ(1⊗ a) = δ⊗a=δ⊗(ff−1a) = (δf)⊗(f−1a) = 1 ⊗(δf)(f−1a) = 1 ⊗(δa)] that the isomorphisms above at different pglue together and give a global isomorphism Im OX,1,1≃OX. This example suggests studying the comparison between DR(Im E,k,k′), k, k′≫0, and ICX(L) in theorem (1.6), independent of the fact that ρE,k and ρE∗,k′are isomorphisms or not. 2 Bernstein-Sato polynomials for cyclic integrable logarithmic connections In the situation of (1.4), let us assume that Eis a cyclic V0-module generated by an element e∈E. The following result is proved in [6, prop. (3.2.3)]. (2.1) Proposition. Under the above conditions, the polynomial bE,p(s)coincides with the Bernstein-Sato polynomial be(s)of ewith respect to f, where eis considered to be an element of the holonomic D-module E[f−1](cf. [12]). (2.2) Let Θf,s ⊂D[s] be the set of operators in annD[s]fsof total order (in s and in the derivatives) ≤1. The elements of Θf,s are of the form δ−αs with δ∈DerC(O), α∈Oand δ(f) = αf. In particular Θf,s ⊂ V0[s]. The O-linear map δ∈Der(log D)p7→ δ−δ(f) fs∈Θf,s 7 is an isomorphism of Lie-Rinehart algebras over (C,O) and extends to a unique ring isomorphism Φ : V0[s]−→ V0[s] with Φ(s) = sand Φ(a) = afor all a∈O. Let us note that Φ−1(δ) = δ+δ(f) fsfor each δ∈Der(log D)p. It is clear that E[s]fsis a sub-V0[s]-module of E[s, f−1]fsand that for any P∈ V0[s] and any e′∈E[s], the following relation holds (Pe′)fs= Φ(P)(e′fs).(5) (2.3) Proposition. Under the above conditions, the following relation holds annV0[s](efs) = V0[s]·Φ (annV0e). Proof. The inclusion ⊃comes from (5). For the other inclusion, let Q∈ annV0[s](efs) and let us write Φ−1(Q) = Pd i=1 Pisiwith Pi∈ V0. We have 0 = Q(efs) = Φ−1(Q)efs= d X i=1 (Pie)si!fs and then Pi∈annV0e. Therefore Q= Φ d X i=1 Pisi!= d X i=1 Φ(Pi)si∈ V0[s]·Φ (annV0e). Q.E.D. (2.4) Proposition. Under the above conditions, if Dis a locally quasihomogeneous free divisor, then annD[s](efs) = D[s]·annV0[s](efs). Proof. From (5) we know that E[s]fs=V0[s]·(efs), and from [6, cor. (3.1.2)] we know that the morphism ρE,s :P⊗(e′fs)∈D[s]⊗V0[s]E[s]fs7→ P(e′fs)∈D[s]·(E[s]fs) = D[s]·(efs) is an isomorphism of left D[s]-modules. Therefore annD[s](efs) = D[s]·annV0[s](efs). Q.E.D. (2.5) Corollary. Under the above conditions, if Dis a locally quasi-homogeneous free divisor, then annD[s](efs) = D[s]·Φ (annV0e). Proof. It follows from propositions (2.3) and (2.4). Q.E.D. 8 (2.6) Remark. Theorems (1.5) and (1.6), proposition (2.4) and corollary (2.5) remain true if we only assume that our divisor Dis Koszul free and of commutative linear type, i.e. its jacobian ideal is of linear type (see [6, §3]). (2.7) Remark. As we shall see in sections 3 and 4, theorem (1.6), proposition (2.1) and corollary (2.5) provide an effective method of computing the intersection DX-module corresponding to ICX(L) in terms of the ILC E, at least if Dis a locally quasi-homogeneous free divisor, or more generally, if Dis Koszul free and of commutative linear type (see remark (2.6)). (2.8) Remark. In the particular case of E=OXand E=O, corollary (2.5) says that annD[s](fs) = D[s]·(δ1−α1s,...,δn−αns), where δ1,...,δnis a local basis of Der(log D)pand δi(f) = αif(see corollary 5.8, (b) in [4]). (2.9) Example. Let us suppose that D⊂Xis a non-necessarily free divisor and let f= 0 be a reduced local equation of Dat a point p∈D. Let {δ1,...,δm} a system of generators of Der(log D)pand let us write δi(f) = αif. Let us call ann(1) D[s](fs) the ideal of D[s] generated by Θf,s (see (2.2)): ann(1) D[s](fs) = D[s]·(δ1−α1s, . . . , δm−αms)⊂annD[s](fs). The Bernstein functional equation for f b(s)fs=P(s)fs+1 means that the operator b(s)−P(s)fbelongs to the annihilator of fsover D[s]. Then, an explicit knowledge of the ideal annD[s](fs) allows us to find b(s) by computing the ideal C[s]∩D[s]·f+ annD[s](fs), (see [25]). However, the ideal annD[s](fs) is in general difficult to compute. When Dis a locally quasi-homogeneous free divisor, or more generally, a divisor of differential linear type ([6], def. (1.4.5) ), annD[s](fs) = ann(1) D[s](fs) and the computation of b(s) is in principle easier. But there are other examples where the Bernstein polynomial b(s) belongs to C[s]∩D[s]·f+ ann(1) D[s](fs) even if annD[s](fs)6= ann(1) D[s](fs). For instance, when X=C3and f= x1x2(x1+x2)(x1+x2x3) (see example 6.2 in [4]) or in any of the examples in page 445 of [9]. In all this examples the divisor is free and satisfies the logarithmic comparison theorem. 9 holds, then Q∈Kν,µ. Actually, by using the equality [Q, g1] = 4Qand the fact that σ(Q) = σ(Q0) and σ(g1) = σ(δ1) also form a regular sequence in Gr W2, condition (11) implies that Kν,µ =W2(g1, Q), σ Kν,µ= (σ(δ1), σ(Q0)). On the other hand, since σ(Q0) is not contained in the ideal (x1, x2), we finally deduce the following result: If parameters ν, µ = (λ, m, n) satisfy conditons (i)-(iv) and (11), then the conormal of the origin T∗ 0(X) does not appear as an irreducible component of the characteristic variety of Im θν,µ =W2/Kν,µ, and consequently Ch(ICX(Lν,µ)) = Ch W2/Kν,µ={σ(δ1) = σ(Q0) = 0}=T∗ X(X)∪T∗ D(X). The existence of such an example has been suggested by [21], example (3.4), but the question on the values of the parameters ν, µ for which the local system Lν,µ is irreducible will be treated elsewhere. If condition (11) does not hold, it is not clear that there exists a general expression for a system of generators of Kν,µ as before. (4.2) Remark. 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