A computational approach to free divisors and logarithmic D-modules

A computational approach to free divisors and logarithmic D-modules J.M. Ucha and F.J. Castro-Jim´enez∗ Abstract We apply algorithmic techniques for comparing some D-modules associated to a kind of divisors in Cn. Introduction We will work on some (analytic) D-modules associated to a germ of holomorphic function on Cn. We propose four natural questions on the comparison of these D-modules. Although our general settings are in the analytic category, we prove that if the input is a complex polynomial in nvariables then the associated objects are computable. Let us denote by O=OCnthe sheaf of holomorphic functions on X=Cn. Consider a point x∈Cn. Denote by Der(Ox) the Ox-module of C-derivations of Ox(the elements in Der(Ox) are called vector fields). Let D⊂Xbe a divisor and x∈D. A vector field δ∈Der(Ox) is said to be logarithmic with respect to Dif δ(f) = af for some a∈ Ox, where fis a local (reduced) equation of the germ (D, x)⊂(Cn, x). The Ox-module of logarithmic vector fields (or logarithmic derivations) is denoted by Der(log D)x. This yields a O-module sheaf denoted by Der(log D). Definition 0.1. ([13]) The divisor Dis said to be free at the point x∈Dif the Ox-module Der(log D)xis free. The divisor Dis called free if it is free at each point x∈D. Smooth divisors and normal crossing divisors are free. By [13] any reduced germ of plane curve D⊂C2is a free divisor. By Saito’s criterium [13], D≡(f= 0) ⊂Cnis free at a point xif and only if there exist nvector fields δi=Pn j=1 aij∂j,i= 1, . . . , n, such that det(aij) = uf where uis a unit in Ox. Here ∂jis the partial derivative ∂ ∂xjand aij is a holomorphic function in Ox. ∗Partially supported by DGESIC and FQM-218. 88 1 Logarithmic D-modules Let us denote by D=DXthe sheaf (of rings) of linear differential operators with holomorphic coefficients on X=Cn. A local section Pof D(i.e. a linear differential operator) is a finite sum P=Pαaα∂α where α= (α1, . . . , αn)∈Nn,aαis a local section of Oand ∂= (∂1, . . . , ∂n) with ∂i=∂ ∂xi in some local chart. For any divisor D⊂Cnwe denote by O[?D] the sheaf of meromorphic functions with poles along D. It follows from the results of Bernstein-Bj¨ork ([1], [2]) on the existence of the b-function for each local equation fof D, that O[?D] is a left coherent D-module. Kashiwara proved that the dimension of its characteristic variety has dimension nand then that O[?D] is holonomic, [11]. We follow [3] and [4] to define two D-modules associated to any divisor D. We consider first the (left) ideal Ilog D⊂ D generated by the logarithmic vector fields Der(log D) (see 0.1). On the other hand, in [15] (see also [9]) L. Narv´aez suggested the study of the (left) ideal in D, denoted by e Ilog D, and generated by the set {δ+a|δ∈Ilog Dand δ(f) = af}. Let us write f Mlog D=D/e Ilog D. There exists a natural morphism φD:f Mlog D→ O[?D] defined by φD(P) = P(1/f) where Pdenotes the class of the operator P∈ D modulo e Ilog. The image of φDis D1 f, i.e. the D-submodule of O[?D] generated by 1/f. As a natural question we ask for the relationship between Mlog D,f Mlog Dand O[?D]. We will listed below some partial answers to this question. When Dis defined by a polynomial f∈C[x1, . . . , xn], then the ideals Ilog Dand e Ilog D are computable. We simply compute a system of generators (over the polynomial ring C[x1. . . , xn]) of syzygies of (f1, . . . , fn, f) where fi=∂f ∂xi. In this case both ideals Ilog D and e Ilog Dare considered in the Weyl algebra An(C), and this is enough because the inclusion An(C)⊂ D is flat. We say that a free divisor Dis of Spencer type if the complex D ⊗O∧•Der(log D)→Mlog D→0 (introduced in [4]) is a (locally) free resolution of Mlog Dand if this last D-module is holonomic. Theorem 1.1. [10] Suppose Dis of Spencer type. Then (Mlog D)∗≃f Mlog D. Here (Mlog D)∗is the dual D-module of Mlog D. Theorem 1.2. ([15, 9]) In dimension 2. The morphism φDis an isomorphism if and only if Dis a quasi-homogeneous plane curve. Theorem 1.3. [10] Suppose the divisor D⊂Cnis free and locally quasi-homogeneous. Then the morphism φDis an isomorphism (so, f Mlog Dand O[?D]are isomorphic as Dmodules). 89 2 Open problems trough a worked example Here we will explicitly compare logarithmic D-modules and O[?D]. We will treat here the divisor D⊂C3whose local equation at (0,0,0) is given by f= 0 with f=x(x2−y3)(x2−zy3). This divisor is (globally) free and δ1, δ2, δ3form a (global) basis of Der(logD), where δ1=3 2x∂x+y∂y δ2= (y3z−x2)∂z δ3= (−1 2xy2)∂x−1 3x2∂y+ (y2z2−y2z)∂z, whose coefficients verify that       3 2x y 0 0 0 y3z−x2 −1 2xy2−1 3x2y2z2−y2z      =−1 2f. This last equality implies, by Saito’s criterium [13], that Der(log D) is a free module (of rank 3) over the ring Oof convergent power series in 3 variables. In fact, the global Der(log D) is in this case a free module (of rank 3) over the ring Rof polynomials in three variables and we have shown a basis. However, if a global divisor Din C3(i.e. a divisor D defined by a polynomial f) is free at each point of C3, it is locally free and then free over the polynomial ring, by Quillen-Suslin theorem. So, it is possible to find a global basis of Der(log f) as a R-module, using for example the Logar-Sturmfels algorithm. To obtain (Mlog D)∗≃f Mlog Dwe have to follow two steps: •Step 1: Check if Mlog Dis holonomic. This computation could be made with [14]. The interest of this question is evident: if Mlog Dis not holonomic, the computation of its dual could not be managed as we will do. •Step 2: Compute a free resolution of Mlog Dwith Gr¨obner basis computation of syzygies. Check if Dis of Spencer type. If this happens then duality holds by [10]. We return to our example. It verifies these properties. The module Syz(δ1, δ2, δ3) is generated by the syzygies obtained from the commutators [δi, δj]. We have Syz(δ1, δ2δ3) = hs12,s13,s23iwhere s12 = (−δ2, δ1−3,0) s13 = (−δ3,0, δ1−2) s12 = (0,−δ3−y2z, δ2). On the other hand, the module Syz(s12,s13,s23) is generated by the element r: r= (−y2z2∂z+y2z∂z+1 2xy2∂x−y2z+1 3x2∂y, y3z∂z−x2∂z,−y∂y−3 2x∂x+ 5). 90 This is the element required to have the Spencer type resolution so, as we have said, duality holds because our divisor f= 0 is of Spencer type. As another consequence of this fact, we have in this example, by [4] Ω•(log D)≃Sol(Mlog D) here Ω•(log D) is the complex of logarithmic differential forms with respect to fand Sol() means the solution complex of the corresponding module. Once duality is proved, a new problem is to obtain the least integer root α0of the b-function because O[?D]≃ D1/fα0[11]. If the least integer root is -1, then O[?D]≃ D · 1/f ≃AnnD(1/f). As in this example e Ilog D=AnnD(1/f) then Logarithmic Comparison Theorem would hold: Ω•[?D]≃DR(O[?D]) ≃DR(f Mlog D)≃ ≃DR((Mlog D)∗)≃Sol(Mlog D)≃Ω•(log D) here DR() means the de Rham complex. The computations of the global b-function of fand AnnD(1/f) could be made with [14] again, that is, using the algorithms of [12]. So, as we have seen, the following problems arise naturally (for a general divisor Din Cn): 1. If Dis free, is Mlog Dholonomic? 2. If Dis free, does Mlog Dadmits a Spencer type resolution? 3. Are there free, not of Spencer type divisors such that duality holds? 4. When the ideals e Ilog Dand AnnD(1/f) coincide? Complexity related problem.- Obtaining the b-function and the annihilator of 1/fα could be a problem of a big complexity. Try the family xp+yq+xyq−1+zxrwhere 4≤p < q ≤r. Help.- When the complexity of obtaining the b-function or AnnD(1/f) is intractable, we have an auxiliary tool to check if O[?D]≃f Mlog D: compute the Extn D(f Mlog D,O) and try to find an argument to assure that is not zero. As Extn D(O[?D],O) = 0 by the regularity of O, this is an effective way to compare this modules. Sometimes the differential equations are not that difficult to solve! ([15]). 91 References [1] Bernstein, I. N., Analytic continuation of generalized functions with respect to a parameter, Functional Anal. and its Applications 6, (1972), p. 273-285 [2] Bj¨ork, J-E. Rings of Differential Operators. North-Holland, Amsterdam 1979. [3] Calder´on-Moreno, F.J. Operadores diferenciales logar´ıtmicos con respecto a un divisor libre. Ph.D. Thesis. June 1997. [4] Calder´on-Moreno, F.J. Logarithmic Differential Operators and Logarithmic De Rham Complexes relative to a Free Divisor. Ann. Sci. E.N.S., 4es´erie, t. 32, 1999, p. 701-714. [5] Calder´on-Moreno F.J., D. Mond, L. Narv´aez-Macarro and F.J. Castro-Jim´enez. Logarithmic Cohomology of the Complement of a Plane Curve. Preprint, University of Warwick, 3/1999. [6] Calder´on-Moreno, F.J. and Narv´aez-Macarro, L. Locally quasi-homogeneous free divisors are Koszul free. Prepub. Fac. Matem´aticas, Univ. Sevilla, n. 56; Oct. 1999. [7] Calder´on-Moreno, F.J. and Narv´aez-Macarro, L. The module Dfsfor locally quasihomogeneous free divisors. Prepub. Dept. ´ Algebra, Univ. Sevilla, n. 4, May, 2000. [8] Castro-Jim´enez, F.J., Mond, D. and Narv´aez-Macarro, L. Cohomology of the complement of a free divisor. Trans. Amer. Math. Soc. 348 (1996), no. 8, 3037–3049. [9] Castro-Jim´enez, F.J., Ucha-Enr´ıquez, J.M. Explicit comparison theorems for Dmodules. To appear in Journal of Symbolic Computation. [10] Castro-Jim´enez, F.J., Ucha-Enr´ıquez, J.M. Free divisors and duality for D-modules. ArXiv.math.AG.0103085; March 14, 2001 [11] Kashiwara, M. On the holonomic systems of linear differential equations II, Invent. Math., 49, 1978, 2, 121–135 [12] Oaku, T. Algorithms for the b-function and D-modules associated with a polynomial. J. of Pure and Appl. Alg., 117 & 118:495-518, 1997. [13] Saito, K. Theory of logarithmic differential forms and logarithmic vector fields. J. Fac Sci. Univ. Tokyo 27:256-291, 1980. [14] Takayama, N. Kan: a system for computation in Algebraic Analysis. Source code available for Unix computers from ftp.math.kobe-u.ac.jp, 1991. [15] Ucha-Enr´ıquez, J.M. M´etodos constructivos en ´algebras de operadores diferenciales. Tesis Doctoral, Universidad de Sevilla, 1999. F.J. Castro-Jim´enez and J.M. Ucha; Depto. de ´ Algebra, University of Sevilla; Apdo. 1160, E-41080 SEVILLA (Spain); e-mail:{castro, ucha}@algebra.us.es 92