A computational approach to free divisors and logarithmic D-modules

A compu a ional app oach o ee di iso s and loga i hmic D-modules J.M. Ucha and F.J. Cas o-Jim´enez∗ Abs ac We apply algo i hmic echniques o compa ing some D-modules associa ed o a kind o di iso s in Cn. In oduc ion We will wo k on some (analy ic) D-modules associa ed o a ge m o holomo phic unc ion on Cn. We p opose ou na u al ques ions on he compa ison o hese D-modules. Al hough ou gene al se ings a e in he analy ic ca ego y, we p o e ha i he inpu is a complex polynomial in n a iables hen he associa ed objec s a e compu able. Le us deno e by O=OCn he shea o holomo phic unc ions on X=Cn. Conside a poin x∈Cn. Deno e by De (Ox) he Ox-module o C-de i a ions o Ox( he elemen s in De (Ox) a e called ec o ields). Le D⊂Xbe a di iso and x∈D. A ec o ield δ∈De (Ox) is said o be lo- ga i hmic wi h espec o Di δ( ) = a o some a∈ Ox, whe e is a local ( educed) equa ion o he ge m (D, x)⊂(Cn, x). The Ox-module o loga i hmic ec o ields (o loga i hmic de i a ions) is deno ed by De (log D)x. This yields a O-module shea deno ed by De (log D). De ini ion 0.1. ([13]) The di iso Dis said o be ee a he poin x∈Di he Ox-module De (log D)xis ee. The di iso Dis called ee i i is ee a each poin x∈D. Smoo h di iso s and no mal c ossing di iso s a e ee. By [13] any educed ge m o plane cu e D⊂C2is a ee di iso . By Sai o’s c i e ium [13], D≡( = 0) ⊂Cnis ee a a poin xi and only i he e exis n ec o ields δi=Pn j=1 aij∂j,i= 1, . . . , n, such ha de (aij) = u whe e uis a uni in Ox. He e ∂jis he pa ial de i a i e ∂ ∂xjand aij is a holomo phic unc ion in Ox. ∗Pa ially suppo ed by DGESIC and FQM-218. 88 1 Loga i hmic D-modules Le us deno e by D=DX he shea (o ings) o linea di e en ial ope a o s wi h holomo p- hic coe icien s on X=Cn. A local sec ion Po D(i.e. a linea di e en ial ope a o ) is a ini e sum P=Pαaα∂α whe e α= (α1, . . . , αn)∈Nn,aαis a local sec ion o Oand ∂= (∂1, . . . , ∂n) wi h ∂i=∂ ∂xi in some local cha . Fo any di iso D⊂Cnwe deno e by O[?D] he shea o me omo phic unc ions wi h poles along D. I ollows om he esul s o Be ns ein-Bj¨o k ([1], [2]) on he exis ence o he b- unc ion o each local equa ion o D, ha O[?D] is a le cohe en D-module. Kashiwa a p o ed ha he dimension o i s cha ac e is ic a ie y has dimension nand hen ha O[?D] is holonomic, [11]. We ollow [3] and [4] o de ine wo D-modules associa ed o any di iso D. We conside i s he (le ) ideal Ilog D⊂ D gene a ed by he loga i hmic ec o ields De (log D) (see 0.1). On he o he hand, in [15] (see also [9]) L. Na ´aez sugges ed he s udy o he (le ) ideal in D, deno ed by e Ilog D, and gene a ed by he se {δ+a|δ∈Ilog Dand δ( ) = a }. Le us w i e Mlog D=D/e Ilog D. The e exis s a na u al mo phism φD: Mlog D→ O[?D] de ined by φD(P) = P(1/ ) whe e Pdeno es he class o he ope a o P∈ D modulo e Ilog. The image o φDis D1 , i.e. he D-submodule o O[?D] gene a ed by 1/ . As a na u al ques ion we ask o he ela ionship be ween Mlog D, Mlog Dand O[?D]. We will lis ed below some pa ial answe s o his ques ion. When Dis de ined by a polynomial ∈C[x1, . . . , xn], hen he ideals Ilog Dand e Ilog D a e compu able. We simply compu e a sys em o gene a o s (o e he polynomial ing C[x1. . . , xn]) o syzygies o ( 1, . . . , n, ) whe e i=∂ ∂xi. In his case bo h ideals Ilog D and e Ilog Da e conside ed in he Weyl algeb a An(C), and his is enough because he inclusion An(C)⊂ D is la . We say ha a ee di iso Dis o Spence ype i he complex D ⊗O∧•De (log D)→Mlog D→0 (in oduced in [4]) is a (locally) ee esolu ion o Mlog Dand i his las D-module is holo- nomic. Theo em 1.1. [10] Suppose Dis o Spence ype. Then (Mlog D)∗≃ Mlog D. He e (Mlog D)∗is he dual D-module o Mlog D. Theo em 1.2. ([15, 9]) In dimension 2. The mo phism φDis an isomo phism i and only i Dis a quasi-homogeneous plane cu e. Theo em 1.3. [10] Suppose he di iso D⊂Cnis ee and locally quasi-homogeneous. Then he mo phism φDis an isomo phism (so, Mlog Dand O[?D]a e isomo phic as D- modules). 89 2 Open p oblems ough a wo ked example He e we will explici ly compa e loga i hmic D-modules and O[?D]. We will ea he e he di iso D⊂C3whose local equa ion a (0,0,0) is gi en by = 0 wi h =x(x2−y3)(x2−zy3). This di iso is (globally) ee and δ1, δ2, δ3 o m a (global) basis o De (logD), whe e δ1=3 2x∂x+y∂y δ2= (y3z−x2)∂z δ3= (−1 2xy2)∂x−1 3x2∂y+ (y2z2−y2z)∂z, whose coe icien s e i y ha       3 2x y 0 0 0 y3z−x2 −1 2xy2−1 3x2y2z2−y2z      =−1 2 . This las equali y implies, by Sai o’s c i e ium [13], ha De (log D) is a ee module (o ank 3) o e he ing Oo con e gen powe se ies in 3 a iables. In ac , he global De (log D) is in his case a ee module (o ank 3) o e he ing Ro polynomials in h ee a iables and we ha e shown a basis. Howe e , i a global di iso Din C3(i.e. a di iso D de ined by a polynomial ) is ee a each poin o C3, i is locally ee and hen ee o e he polynomial ing, by Quillen-Suslin heo em. So, i is possible o ind a global basis o De (log ) as a R-module, using o example he Loga -S u m els algo i hm. To ob ain (Mlog D)∗≃ Mlog Dwe ha e o ollow wo s eps: •S ep 1: Check i Mlog Dis holonomic. This compu a ion could be made wi h [14]. The in e es o his ques ion is e iden : i Mlog Dis no holonomic, he compu a ion o i s dual could no be managed as we will do. •S ep 2: Compu e a ee esolu ion o Mlog Dwi h G ¨obne basis compu a ion o syzy- gies. Check i Dis o Spence ype. I his happens hen duali y holds by [10]. We e u n o ou example. I e i ies hese p ope ies. The module Syz(δ1, δ2, δ3) is gene a ed by he syzygies ob ained om he commu a o s [δi, δj]. We ha e Syz(δ1, δ2δ3) = hs12,s13,s23iwhe e s12 = (−δ2, δ1−3,0) s13 = (−δ3,0, δ1−2) s12 = (0,−δ3−y2z, δ2). On he o he hand, he module Syz(s12,s13,s23) is gene a ed by he elemen : = (−y2z2∂z+y2z∂z+1 2xy2∂x−y2z+1 3x2∂y, y3z∂z−x2∂z,−y∂y−3 2x∂x+ 5). 90 This is he elemen equi ed o ha e he Spence ype esolu ion so, as we ha e said, duali y holds because ou di iso = 0 is o Spence ype. As ano he consequence o his ac , we ha e in his example, by [4] Ω•(log D)≃Sol(Mlog D) he e Ω•(log D) is he complex o loga i hmic di e en ial o ms wi h espec o and Sol() means he solu ion complex o he co esponding module. Once duali y is p o ed, a new p oblem is o ob ain he leas in ege oo α0o he b- unc ion because O[?D]≃ D1/ α0[11]. I he leas in ege oo is -1, hen O[?D]≃ D · 1/ ≃AnnD(1/ ). As in his example e Ilog D=AnnD(1/ ) hen Loga i hmic Compa ison Theo em would hold: Ω•[?D]≃DR(O[?D]) ≃DR( Mlog D)≃ ≃DR((Mlog D)∗)≃Sol(Mlog D)≃Ω•(log D) he e DR() means he de Rham complex. The compu a ions o he global b- unc ion o and AnnD(1/ ) could be made wi h [14] again, ha is, using he algo i hms o [12]. So, as we ha e seen, he ollowing p oblems a ise na u ally ( o a gene al di iso Din Cn): 1. I Dis ee, is Mlog Dholonomic? 2. I Dis ee, does Mlog Dadmi s a Spence ype esolu ion? 3. A e he e ee, no o Spence ype di iso s such ha duali y holds? 4. When he ideals e Ilog Dand AnnD(1/ ) coincide? Complexi y ela ed p oblem.- Ob aining he b- unc ion and he annihila o o 1/ α could be a p oblem o a big complexi y. T y he amily xp+yq+xyq−1+zx whe e 4≤p < q ≤ . Help.- When he complexi y o ob aining he b- unc ion o AnnD(1/ ) is in ac able, we ha e an auxilia y ool o check i O[?D]≃ Mlog D: compu e he Ex n D( Mlog D,O) and y o ind an a gumen o assu e ha is no ze o. As Ex n D(O[?D],O) = 0 by he egula i y o O, his is an e ec i e way o compa e his modules. Some imes he di e en ial equa ions a e no ha di icul o sol e! ([15]). 91 Re e ences [1] Be ns ein, I. N., Analy ic con inua ion o gene alized unc ions wi h espec o a pa a- me e , Func ional Anal. and i s Applica ions 6, (1972), p. 273-285 [2] Bj¨o k, J-E. Rings o Di e en ial Ope a o s. No h-Holland, Ams e dam 1979. [3] Calde ´on-Mo eno, F.J. Ope ado es di e enciales loga ´ı micos con espec o a un di iso lib e. Ph.D. Thesis. June 1997. [4] Calde ´on-Mo eno, F.J. Loga i hmic Di e en ial Ope a o s and Loga i hmic De Rham Complexes ela i e o a F ee Di iso . Ann. Sci. E.N.S., 4es´e ie, . 32, 1999, p. 701-714. [5] Calde ´on-Mo eno F.J., D. Mond, L. Na ´aez-Maca o and F.J. Cas o-Jim´enez. Lo- ga i hmic Cohomology o he Complemen o a Plane Cu e. P ep in , Uni e si y o Wa wick, 3/1999. [6] Calde ´on-Mo eno, F.J. and Na ´aez-Maca o, L. Locally quasi-homogeneous ee di i- so s a e Koszul ee. P epub. Fac. Ma em´a icas, Uni . Se illa, n. 56; Oc . 1999. [7] Calde ´on-Mo eno, F.J. and Na ´aez-Maca o, L. The module D s o locally quasi- homogeneous ee di iso s. P epub. Dep . ´ Algeb a, Uni . Se illa, n. 4, May, 2000. [8] Cas o-Jim´enez, F.J., Mond, D. and Na ´aez-Maca o, L. Cohomology o he comple- men o a ee di iso . T ans. Ame . Ma h. Soc. 348 (1996), no. 8, 3037–3049. [9] Cas o-Jim´enez, F.J., Ucha-En ´ıquez, J.M. Explici compa ison heo ems o D- modules. To appea in Jou nal o Symbolic Compu a ion. [10] Cas o-Jim´enez, F.J., Ucha-En ´ıquez, J.M. F ee di iso s and duali y o D-modules. A Xi .ma h.AG.0103085; Ma ch 14, 2001 [11] Kashiwa a, M. On he holonomic sys ems o linea di e en ial equa ions II, In en . Ma h., 49, 1978, 2, 121–135 [12] Oaku, T. Algo i hms o he b- unc ion and D-modules associa ed wi h a polynomial. J. o Pu e and Appl. Alg., 117 & 118:495-518, 1997. [13] Sai o, K. Theo y o loga i hmic di e en ial o ms and loga i hmic ec o ields. J. Fac Sci. Uni . Tokyo 27:256-291, 1980. [14] Takayama, N. Kan: a sys em o compu a ion in Algeb aic Analysis. Sou ce code a ailable o Unix compu e s om p.ma h.kobe-u.ac.jp, 1991. [15] Ucha-En ´ıquez, J.M. M´e odos cons uc i os en ´algeb as de ope ado es di e enciales. Tesis Doc o al, Uni e sidad de Se illa, 1999. F.J. Cas o-Jim´enez and J.M. Ucha; Dep o. de ´ Algeb a, Uni e si y o Se illa; Apdo. 1160, E-41080 SEVI- LLA (Spain); e-mail:{cas o, ucha}@algeb a.us.es 92