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.
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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).
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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).
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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]).
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Re e ences
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[14] Takayama, N. Kan: a sys em o compu a ion in Algeb aic Analysis. Sou ce code
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[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
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