a Xi :ma h/0204162 1 [ma h.AG] 12 Ap 2002
A Compu a ional App oach o he D-module o
Me omo phic Func ions
F.J. Cas o-Jim´enez and J.M. Ucha
Sep embe , 2001
Abs ac
Le Dbe a di iso in Cn. We p esen me hods o compa e he D-module o he
me omo phic unc ions O[∗D] o some na u al app oxima ions. We show how he ana-
ly ic case can be ea ed wi h compu a ions in he Weyl algeb a.
1 In oduc ion
Le us deno e by O=OCn he shea o holomo phic unc ions on X=Cn. Conside a
poin p∈Cn.De (Op) is he Op-module o C-de i a ions o Op. The elemen s in De (Op)
a e called ec o ields.
Le D⊂Xbe a di iso (i.e. a hype su ace) and p∈D. A ec o ield δ∈De (Op) is
said o be loga i hmic wi h espec o Di δ( ) = a o some a∈ Op, whe e is a local
( educed) equa ion o he ge m (D, p)⊂(Cn, p). The Op-module o loga i hmic ec o
ields (o loga i hmic de i a ions) is deno ed by De (log D)p. This yields an O-module
shea deno ed by De (log D) (see [15]).
Le us deno e by D=DX he shea (o ings) o linea di e en ial ope a o s wi h
holomo phic 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∂αwhe e α= (α1,... ,αn)∈Nn,aαis a local sec ion
o Oand ∂= (∂1,... ,∂n) wi h ∂i=∂
∂xiin 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 is nand hen ha O[⋆D]
is holonomic, [12].
We will conside some D-modules associa ed o any di iso D:
•The (le ) ideal Ilog D⊂ D gene a ed by he loga i hmic ec o ields De (log D).
•The (le ) ideal e
Ilog D⊂ D gene a ed by he se {δ+a|δ∈Ilog Dand δ( ) = a }.
Mo e gene ally, he ideals e
I(k) log Dgene a ed by he se
{δ+ka |δ∈Ilog Dand δ( ) = a }
1
•The modules Mlog D=D/Ilog D,
Mlog D=D/e
Ilog Dand mo e gene ally
M(k) log D=
D/e
I(k) log D.
The inclusion e
I(k) log D⊂AnnD(1/ k) yields o a na u al mo phism φk
D:
M(k) log D→
O[⋆D] de ined by φk
D(P) = P(1/ k) whe e Pdeno es he class o he ope a o P∈ D
modulo e
I(k) log D. The image o φk
Dis D1
k, i.e. he D-submodule o O[⋆D] gene a ed by
1/ k.
Conside ing he gene al ideals e
I(k) log Dis a sugges ion o P o . Tajima. The poin is he
well known chain o inclusions
D · −1⊂ D · −2⊂ · · · ⊂ D · k=D · k−1=···=O[⋆D],
whe e kis leas in ege oo o he b- unc ion.
We a e in e es ed in he ge m (D, p)⊂(Cn, p) o a ixed poin p∈D. So we will wo k
in he ing o ge ms o linea di e en ial ope a o s Dp. We suppose ha p= 0 ∈Cnand
om now on, we will deno e D=D0. In his con ex we will use he Weyl algeb a An(C)
as a sub ing o D.
Unde a compu a ional poin o iew he di iso Dwill be de ined by a polynomial
∈C[x1,... ,xn]. The b- unc ion o is compu able by [14] and we gi e a di ec me hod
o p esen O[∗D]. I he calcula ion is in ac able –as some imes happens in he examples–
we also p esen an indi ec me hod o deduce ha O[∗D] and he modules
M(k) log Ddo no
coincide. The me hod is s ongly based in he ollowing esul o [13]:
Theo em 1.1. The es ic ion o Do he shea Ex i
D(O[∗D],O)is ze o o i≥0.
Mo e p ecisely, we p o e ha , unde ce ain algo i hmic condi ions, some cohomology
g oups a e no ze o. The in e es o his second me hod has been es ed in [18], [9] and [10].
We poin ou how he algo i hms p esen ed in [17] – ha only calcula e cohomology g oups
in he algeb aic case– could also be use ul in some analy ic si ua ions.
I is impo an o unde line ha ou me hods manage he analy ic case. As he inclusion
An(C)⊂ D is la , he compu a ion o syzygies and ee esolu ions in he Weyl algeb a
yields o he analogous compu a ions in D.
2 Compa ison algo i hms
We p opose in his sec ion wo me hods o compa e he loga i hmic modules p esen ed
abo e. I is impo an o ema k ha he compu a ion o he analy ic De (log D) can
be made o a di iso Di i s local equa ion is a polynomial ∈C[x1,... ,xn] . Simply
compu e, using G ¨obne basis, a sys em o gene a o s o ( 1, . . . , n, ) whe e i=∂
∂xi
because he inclusion o he Weyl algeb a in Dis la .
2
2.1 Di ec compa ison
The i s me hod is comple e bu needs he calcula ion o he b- unc ion.
Expe imen al e idences show ha i he di iso is no locally Eule homogeneous (i.e.
he e is no δ∈De (log D) such ha δ( ) = ) he b- unc ion is ha d o compu e. Mo e
p ecisely, he p oblem seems o be he calcula ion o AnnD[s]( s) and he use o ce ain
elimina ion o de s du ing he calcula ion o G ¨obne basis (he e D[s] s ands o he poly-
nomial ing, wi h he inde e mina e scommu ing wi h D). Ne e heless, he me hod is
applicable o all he Eule homogeneous di iso s we ha e conside ed (and o some no Eule
homogeneous).
Algo i hm 2.1. INPUT: A local equa ion = 0 o a di iso D;
1. Compu e he b- unc ion o . Le −α0be he leas in ege oo .
2. Compu e he ideal AnnD(1/ α0).
3. Compu e a se o gene a o s {s1,... ,s }o Syz( 1,... , n, ). The ideal e
I(α0) log Dis
gene a ed by he elemen s
sj





∂1
.
.
.
∂n
−α0




∈ D.
4. Compa e AnnD(1/ α0) and e
I(α0) log D.
OUTPUT: O[∗D]≃
M(α0) log D⇔AnnD(1/ α0) = e
I(α0) log D.
The co ec ness o he algo i hm is ob ious as
O[∗D]≃ D 1
α0≃ D/AnnD(1/ α0).
2.2 Indi ec deduc ion: a su icien condi ion
This second me hod is an al e na i e way when you can no ob ain he b- unc ion. In he
wo s case, i only needs he compu a ion o a ee esolu ion o
M(α) log D o any in ege
α≥1. Mo e p ecisely, he algo i hm looks o a echnical condi ion in some s ep o he ee
esolu ion. In many examples, i is enough o compu e only he i s syzygies1.
De ini ion 2.2. I
0→ D sϕs
−→ · · · → D 2ϕ2
−→ D 1ϕ1
−→ D π
−→ M→0
is a ee esolu ion o a module M, we will say ha he Successi e Ma ices Condi ion
(SMC) holds a le el ii he wo succesi e mo phisms ϕi, ϕi+1 ha e ma ices e i ying:
1Taking in o accoun ha compu ing a comple e ee esolu ion can be a p oblem o g ea complexi y,
his op ion is e y in e es ing.
3
1. The ma ix o ϕi+1 has no pa in Oin some column j, i.e. all he elemen s in he
j- h column a e in he (le ) ideal gene a ed by ∂1,... ,∂n.
2. The ma ix o ϕihas no cons an s in he in he j- h ow. Tha is, o each Pin he
j- h ow P(1) is a unc ion hwi h h(0) = 0.
Algo i hm 2.3. INPUT: A local equa ion = 0 o a di iso D;
1. Compu e a se o gene a o s {s1,... ,s }o Syz( 1,... , n, ). The ideal e
I(α) log Dis
gene a ed by he elemen s
sj





∂1
.
.
.
∂n
−α




∈ D.
2. Compu e a ee esolu ion o M=
M(α) log D
0→ D sϕs
−→ · · · → D 2ϕ2
−→ D 1ϕ1
−→ D π
−→ M→0.
OUTPUT: IF SMC holds THEN O[∗D]6=
M(α) log D.
We need a lemma o jus i y he algo i hm. I explains he ole o he SMC. The idea is
ob aining an elemen in Ke ϕi+1 ha is no in Imϕi.
Lemma 2.4. Le Dbe a di iso and
0→ D sϕs
−→ · · · → D 2ϕ2
−→ D 1ϕ1
−→ D π
−→
M(α) log D→0 (∗)
a ee esolu ion o
M(α) log D ha e i ies SMC a le el i. Then
Ex i
D(
M(α) log D,O)6= 0.
P oo . To ob ain he Ex g oups, we ha e o apply he unc o HomD(−,O) o (∗). Using
ha
HomD(D ,O)≃ O
we ob ain he complex
0→ O ϕ
1
−→ O 1ϕ
2
−→ O 2→ · · · ϕ
s
−→ O ϕ
s
−→ O s→0,
whe e ϕ
ideno es he mo phism wi h ma ix he ansposed o ϕi. The de i a i es now ac
na u ally.
Then
Ex i
D(
M(α) log D,O) = Ke ϕ
i+1/Imϕ
i.
I he ma ix o ϕi+1 has no pa in Oin he j- h ow, hen e= (0,... ,1,... ,0) –whe e 1
is in he j- h posi ion– is in Ke ϕ
i+1, as he de i a i es applied o 1 a e ze o.
This elemen can no be in Imϕ
ii he ma ix has no cons an s. Applying he ope a o s
o he ma ix i is no possible o ob ain elemen s o deg ee 0.
4
We ha e he key o s a e he main esul o his sec ion: he co ec ness o 2.3.
P oposi ion 2.5. Le Dbe a di iso wi h a ee esolu ion o
M(α) log D ha e i ies SCM
a some le el. Then O[∗D]6=
M(α) log D.
P oo . E iden om 2.4 and 1.1.
A special case o SCM appea s when you ha e a esolu ion o ype
0→ D nϕn
−→ · · · → D 2ϕ2
−→ D 1ϕ1
−→ D π
−→ M→0
o leng h n. Then
Ex n
D(M, O) = O n
Imϕ
n
,
and SCM means, a le el n, ha you can ind in he ma ix o ϕna ow wi h no cons an s.
Rema k 2.6. O cou se, na u al gene aliza ions o he SCM condi ion has o do wi h inding
explici elemen s in some Ke ϕ
i+1 wi h special p ope ies. I is no ha easy in gene al!
Ne e heless he esul s o [17] can be applied in his si ua ion as ollows:
Ex 0
An(C)(R[∗D], R)6= 0 ⇒Ex 0
D(O[∗D],O)6= 0,
whe e Ris he ing o polynomials and R[∗D] is i s localiza ion wi h espec o he equa ion
o D.
3 Applica ion o he Spence case.
In his sec ion we explain how o apply he su icien condi ion o a special case in which a
ailo ed ee esolu ion is p o ided.
De ini ion 3.1. ([15]) The di iso Dis said o be ee a he poin p∈Di he Op-module
De (log D)pis ee. The di iso Dis called ee i i is ee a each poin p∈D.
Smoo h di iso s and no mal c ossing di iso s a e ee. By [15] any educed ge m o
plane cu e D⊂C2is a ee di iso .
By Sai o’s c i e ium [15], D≡( = 0) ⊂Cnis ee a a poin pi 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
Op. He e ∂jis he pa ial de i a i e ∂
∂xjand aij is a holomo phic unc ion in Op.
De ini ion 3.2. 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.
5

The e a e analogous esolu ions o he amily o modules
M(k) log D.
Fo his amily o di iso s, he solu ion complex Sol(Mlog D) ( ha is, he complex
RHomD(Mlog D,O)) is na u ally quasi-isomo phic o Ω•(log D) (as we poin ed in [10] as
a deduc ion o [4]). On he o he hand, a duali y heo em p o ed in [10] has impo an
consequences compa ing
Mlog Dand O[⋆D], namely
Theo em 3.3. ([18, 9]) In dimension 2, he mo phism φ1
Dis an isomo phism i and only
i Dis a quasi-homogeneous plane cu e.
Theo em 3.4. [10] Suppose he di iso D⊂Cnis ee and locally quasi-homogeneous.
Then he mo phism φ1
Dis an isomo phism (so,
Mlog Dand O[⋆D]a e isomo phic as D-
modules).
The me hods p esen ed in sec ion 2 gi e us compu a ional ools o check he compa ison
be ween
Mlog Dand O[⋆D] .
Rema k 3.5. Once you ha e he duali y o [10], you also ha e a s a egy o s udy he
Loga i hmic Compa ison Theo em (LCT), ha is, he complex Ω•(⋆D) o me omo phic
di e en ial o ms and he complex Ω•(log D) a e quasi isomo phic. You ha e o a el
ound he ollowing chain o isomo phisms:
Ω•(⋆D)≃DR(O[⋆D]) ≃DR(D/AnnD(1/ )) ≃DR(
Mlog D)≃
≃DR((Mlog D)∗)≃Sol(Mlog D)≃Ω•(log D),
whe e o each cohe en D–module Mwe deno e by DR(M) he de Rham complex o M
(see [13]).
Rema k 3.6. The e a e wo in e es ing expe imen al sugges ions:
•We don’ know examples o ee di iso s wi h in ege oo s o hei b- unc ion less
han -1.
•We only know ee di iso s o Spence ype.
Finally, we ha e he ollowing esul :
P oposi ion 3.7. Le D≡( = 0) be a Spence di iso . Le δ1,... ,δnbe a basis o
De (log D)wi h δi=Pn
j=1 aij∂j o 1≤j≤n. I Pn
j=1 ∂j(aij) = 0, hen
Mlog Dand
O[⋆D]a e no isomo phic.
6
P oo . The las ma ix o he Spence ee esolu ion o
Mlog Dis o a e y special ype .
I s elemen s (due o he duali y o mulas o [10]) a e o he o m
δi+
n
X
j=1
∂j(aij).
So, applying lemma 2.4 Ex n
D(
Mlog D,O)6= 0,so
Mlog Dand O[⋆D] a e no isomo phic.
Thus he e is no possible quasi-isomo phism be ween DR(D/AnnD(1/ )) and DR(
Mlog D).
4 Examples
In he ollowing examples, he compu a ion o syzygies among polynomials ha e been made
wi h CoCoA (see [11]).
The compu a ions o syzygies in he Weyl Algeb a, global b- unc ions and ideals o ype
AnnD(1/ α) ha e been made wi h kan/sm1, [16], ha is, using he algo i hms o [14].
4.1 Example 1: D≡(x(x2−y3)(x2−zy3) = 0)
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 .
To begin wi h, we ha e o ollow wo s eps:
•S ep 1: Ve i y ha Mlog Dis holonomic2. 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 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 Dhas a ee esolu ion o Spence ype. I his happens hen duali y
holds by [10].
2This compu a ion could be made wi h [16]
7
The example e i ies hese p ope ies:
1. 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).
2. 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).
This is he elemen equi ed o ha e he Spence ype esolu ion so, as we ha e said,
duali y holds.
We calcula e he b- unc ion o . I s leas in ege oo is -1, so
O[⋆D]≃ D · 1/ ≃AnnD(1/ ).
To inish, we check ha e
Ilog D=AnnD(1/ ).
4.2 Example 2: D≡(x4+y4+z4+x2y2z2= 0) ⊂C3
This is di iso is no ee. A se o gene a o s o
Mlog Dis
δ1= (1/8x2y3z2+y) + (−1/32x3y3z2−1/16xyz4−1/4xy)∂x+
+(−1/32x2y4z2+ 1/8x2z2−1/4y2)∂y+ (−1/4yz)∂z,
δ2= (−1/16x4y4z+z) + (1/64x5y4z+ 1/32x3y2z3−1/4xz)∂x+
+(1/64x4y5z−1/16x4yz −1/4yz)∂y+ (1/8x2y2−1/4z2)∂z,
δ3= (−1/16x4y3z3−1/8x2y5z)+
+(1/64x5y3z3+ 1/32x3y5z+ 1/32x3yz5+ 1/16xy3z3)∂x+
+(1/64x4y4z3+ 1/32x2y6z−1/16x4z3−1/8x2y2z−1/4z3)∂y+
+(1/8x2yz2+ 1/4y3)∂z,
δ4= (−1/16xy4z4+x) + (1/64x2y4z4+ 1/32y2z6+ 1/8y2z2−1/4x2)∂x+
+(1/64xy5z4−1/16xyz4−1/4xy)∂y+ (−1/4xz)∂z,
δ5= (1/8x2y2z+ 1/4z3)∂x+ (−1/8xy2z2−1/4x3)∂z,
δ6= (1/8xy5z2+ 1/4x3y3)+
+(−1/32x2y5z2−1/16x4y3−1/16y3z4−1/8x2yz2−1/4y3)∂x+
+(−1/32xy6z2−1/16x3y4+ 1/8xy2z2+ 1/4x3)∂y.
8
The ee esolu ion is huge, bu anyway compu able wi h kan/sm1. I is o ype
0→ D ϕ3
−→ D ϕ2
−→ Dsϕ1
−→ D π
−→ M→0
To use p oposi ion 2.4 you can check ha he e he elemen s in he las ma ix ϕ3has no
cons an s. So
Ex 3
D(
Mlog D,O) = O/im(ϕ3) 6= 0.
You ha e O[∗D]6=
Mlog D.
4.3 Example 3: D≡(x4+y5+xy4+zx6= 0) ⊂C3
In his example we de ec ha we ha e, in ac , a (non-Eule homogeneous) p oduc ha
is a ee di iso 3. The basis o De (log D) is
δ1= (x6z2+ 5/4x5yz2−2x4z−x2−5/4xy)∂x+
+(5/4x5yz2+ 3/2x4y2z2−5/2x3yz −1/2x2y2z−3/4xy −y2)∂y+
+(2x3z2−2xz)∂z,
δ2= (−4/5x4yz −8/5x3y2z−3/4x2y3z+ 25/4x4z+ 125/16x3yz −xy2−1/4y3+
+125/16xy)∂x+ (−x3y2z−39/20x2y3z−9/10xy4z+ 125/16x3yz + 75/8x2y2z−
−3/4y3+ 1/4x2−5/16xy + 25/4y2)∂y+ (−8/5xyz −6/5y2z+ 25/2xz)∂z,
δ3= (−3/10x3y2z−3/8x2y3z+ 25/8x4z+
+125/32x3yz + 8/25x3−1/8y3+ 125/32xy)∂x+ (−3/8x2y3z−9/20xy4z+
+125/32x3yz + 75/16x2y2z+ 2/5x2y−1/50xy2+ 1/40y3+ 1/8x2−5/32xy+
+25/8y2)∂y+ (−3/5y2z+ 25/4xz + 16/25)∂z.
¿F om he monomial 16/25∂zwe deduce by he Flow Theo em ha Dis a p oduc . I
is o Spence ype and he hi d ma ix can be used o show ha Ex 3
D(
Mlog D,O)6= 0 so
O[∗D]6=
Mlog D.
4.4 Example 4: D≡((x+y)(xz +y)(x4+y5+xy4) = 0) ⊂C3
In his las example he di iso D⊂C3has as a local equa ion a (0,0,0) he o m
= (x+y)(xz +y)(x4+y5+xy4) = 0.
3By he way, his si ua ion is impossible in dimension 2.
9