Bounding the degree of solutions to Pfaff equations

Publicacions Ma em`a iques, Vol. 44 (2000), 593–604
BOUNDING THE DEGREE OF SOLUTIONS TO PFAFF
EQUATIONS
Ma co B unella and Lu´
ıs Gus a o Mendes
Abs ac
We s udy hype su aces o complex p ojec i e mani olds which a e
in a ian by a olia ion, o mo e gene ally which a e solu ions o
a P aff equa ion. We bound hei deg ee using classical esul s on
loga i hmic o ms.
1. In oduc ion
S a ing wi h Poinca ´e[Po] and Painle ´e[Pa], many ma hema icians
conside ed he ollowing p oblem: gi en a olia ion Fon CPno deg ee d
and a hype su ace V⊂CPnin a ian by F, is i possible o bound he
deg ee o Vby a numbe h(d) which depends only on d(and no on F)?
In such a gene ali y, he answe is clea ly nega i e: o example, he
cu e {xp=yq}in CP2has deg ee max{p, q}and i is in a ian by he
olia ion gi en by pydx −qxdy = 0, whose deg ee is one (see also [LN]
o o he in e es ing examples). Howe e se e al posi i e esul s ha e
been ob ained by hose au ho s and, ecen ly, by [CL], [Ca], [Ba], [B ]
and [So]. The philosophy behind hese esul s is ha he ailu e o
an uni o m bound h(d) is due o he exis ence o “bad singula i ies” o
Vo o F. Fo ins ance, one finds he bound h(d)=d+2inCP2,
p o ided ha ei he Vhas only no mal c ossing singula i ies [CL], o F
has only nondic i ical singula i ies along V[Ca]. Ou aim is o a ac
he a en ion o he eade o he ela ion be ween his p oblem and
hese esul s and some basic p ope ies o loga i hmic o ms, disco e ed
by Deligne and Bogomolo [De], [Bo].
We shall wo k in he con ex o P aff equa ions, mo e gene al han
olia ions. Fo ou pu poses, he simples defini ion is he ollowing:
Defini ion 1.1. Gi en a complex mani old Xand a holomo phic line
bundle Non X,aP aff equa ion o codimension p,1≤p≤dimC(X)−1,
is a non i ial global sec ion σo Ωp
X⊗N, whe e Ωp
Xdeno es he shea
o holomo phic p- o ms on X.
594 M. B unella, L. Gus a o Mendes
Le us conside he co esponding gene aliza ion o he no ion o lea
o a olia ion. Gi en a hype su ace V, le iVσdeno e he es ic ion
o σ o V, i.e. he sec ion o Ωp
V⊗N|Vob ained by p ojec ing σ|V ia
(Ωp
X⊗N)|V→Ωp
V⊗N|V.
Defini ion 1.2. Gi en a P aff equa ion σ∈H0(X, Ωp
X⊗N) and a
hype su ace V⊂X, we shall say ha Vis a solu ion o σi iVσ≡0.
Some ema ks on hese defini ions may be use ul. Usually one equi es
ha he ze o se (σ)0o a P aff equa ion σ∈H0(X, Ωp
X⊗N) has
codimension ≥2, bu his is no eally impo an . Anyway, i σ∈
H0(X, Ωp
X⊗N) anishes on a hype su ace Z⊂X, hen we can eplace
σby σ:= σ
h∈H0(X, Ωp
X⊗N), whe e N=N⊗O
X(−Z) and h∈
H0(X, OX(Z)) anishes on Z, and hen σ|Z≡ 0. This di ision o σby h
has no significan consequences, because in he s udy o P aff equa ions
one is mo e in e es ed in he sa u a ed subshea o Ωp
Xgene a ed by σ
han in σi sel , and his subshea is unchanged by he di ision.
A olia ion Fo codimension pgi es ise o a P aff equa ion o codi-
mension p, because locally such a olia ion can be seen as he ke nel o a
holomo phic p- o m; in his case, Nco esponds o he de e minan line
bundle o he ank pno mal shea o F. Bu he con e se is no gene ally
ue, excep when p= dimC(X)−1, because no in eg abili y assump ion
is done. When he P aff equa ion a ises om a olia ion F, Defini ion 1.2
educes o say ha he hype su ace Vis sa u a ed by he lea es o F.
In o de o s a e ou esul in a simple o m, le us suppose now ha
Xis a p ojec i e mani old wi h Pica d g oup Pic(X)=Z, al hough
a mo e gene al ac , wi hou es ic ions on Pic(X), will be ound in
he cou se o he p oo . Le Hbe he posi i e gene a o o Pic(X).
We na u ally define he deg ee o a holomo phic line bundle Las he
in ege d(L) such ha L=d(L)Hin Pic(X). Fo a hype su ace V, he
deg ee d(V) is defined as d(OX(V)). We ecall ha p ojec i e mani olds
wi h Pic(X)=Za e qui e abundan : o ins ance, he classical Noe he -
Le sche z heo em s a es ha a gene ic hype su ace Xin CPno high
deg ee has Pic(X)=Z. We e e o Sec ion 2 o he basic p ope ies o
loga i hmic o ms.
Theo em. Le Xbe a complex p ojec i e mani old wi h Pic(X)=Z
and le σ∈H0(X, Ωp
X⊗N)be a P aff equa ion. Le V⊂Xbe a no mal
c ossing hype su ace, which is solu ion o σ. Then
d(V)≤d(N)
and he inequali y is s ic i Vis smoo h. Mo eo e , i d(V)=d(N) hen
σis gi en by a global closed loga i hmic p- o m on Xwi h poles along V.
Deg ee o Solu ions o P a Equa ions 595
When dimC(X) = 2, we eco e esul s o [CL] and [Ba]. Le us
conside he case when a P aff equa ion on CPna ises om a olia ion
wi h codimCSing(F)≥2 (and codimC(σ)0≥2). One usually defines
he deg ee d(F) as he deg ee o he angency se be ween he (n−p)-di-
mensional lea es o Fand a gene ic linea subspace Π ≃CPp( ema k
ha his angency se is a hype su ace o Π). In Lemma 3.2 we show
ha d(N)=d(F)+p+ 1 and we eco e he esul o [CL]inCP2,
since ou bound becomes
d(V)≤d(F)+p+1.
A he end o he pape we shall discuss o which ex en he no mal
c ossing hypo hesis on Vmay be weakened.
Acknowledgemen s. We hank he hospi ali y o he C.R.M., Ba ce-
lona, whe e his wo k was comple ed. The second au ho is suppo ed
by a g an o “Conseil R´egional de Bou gogne”, F ance, and he hanks
he a en ion o P. Sad and M. G. Soa es.
2. Loga i hmic o ms
In his sec ion we ecall basic ac s on loga i hmic o ms (see, o
ins ance, [Sa]).
Le Xbe a complex mani old and Va hype su ace wi h a mos no -
mal c ossing singula i ies. A loga i hmic p- o m on Xwi h poles along
Vis a me omo phic p- o m ωwi h pola se (ω)∞⊂Vsuch ha ωand
dω ha e a mos simple poles along V. Equi alen ly, i = 0 is a local
educed equa ion o V, hen ω and dω a e holomo phic. Ob iously,
his condi ion is equi alen o ω and d ∧ωbeing holomo phic.
This defini ion can be localized on open se s o X; he e o e we ob-
ain a (cohe en , analy ic, locally ee) shea on X, deno ed Ωp
X(log V).
Rema k ha e e y hing makes sense e en i p=0o p=n= dimC(X),
and Ω0
X(log V)=OX,Ω
n
X(log V)≃Ωn
X⊗O
X(V).
I (z1,...,z
n) is a local coo dina e sys em a ound x=(0,...,0) ∈V
such ha Vis locally exp essed by V={z1·... ·zk=0}, hen
n
p=0 Ωp
X(log V) is locally gene a ed by holomo phic o ms and
{dz1
z1,...,dzk
zk}. Mo e p ecisely, e e y sec ion ωo Ωp
X(log V) can be
locally w i en as
ω=ω0+
k

j=1
ωj∧dzj
zj
,(1)
whe e ω0is a holomo phic p- o m and each ωjis a local sec ion o
Ωp−1
X(log V). Rema k ha he ex e io p oduc o wo loga i hmic o ms
596 M. B unella, L. Gus a o Mendes
is s ill a loga i hmic o m. We also no e ha he exis ence o a decom-
posi ion like (1) is s ongly dependen on he hypo hesis ha Vhas only
no mal c ossing singula i ies.
In pa icula , o e e y j=1,...,k, we can locally decompose ωas
ω=γj+ηj∧dzj
zj
,(2)
whe e γjis a local sec ion o Ωp
X(log V), ηjis a local sec ion o
Ωp−1
X(log V) and mo eo e bo h γjand ηjdo no con ain Vj:= {zj=0}
in hei pola se . The decomposi ion (1), as well as (2), is no unique;
howe e , a simple compu a ion shows ha he es ic ion o ηj o Vjis
in insically defined by ω, i.e. does no depend on he in ol ed choices.
Se ing
Γj:= Vj∩(∪k
i=1,i=jVi),
which is a no mal c ossing hype su ace o Vj, we he e o e ha e a well
defined map
ResVj:ω→ ηj|Vj,
and ηj|Vj∈Ωp−1
Vj(log Γj) is called local esidue o ωalong Vj. Summing
on jand pa ching oge he hese local cons uc ions, we finally ob ain
he esidue map
Res: Ωp
X(log V)→Ωp−1
ˆ
V(log Γ),
whe e ˆ
Vis he no maliza ion o Vand Γ ⊂ˆ
Vis he no mal c ossing
hype su ace induced by Von ˆ
V. No e ha he ke nel o he esidue
map is exac ly Ωp
Xand ha he e is an exac sequence
0→Ωp
X→Ωp
X(log V)→Ωp−1
ˆ
V(log Γ) →0.
The nex lemma was p o ed in [De], as a by-p oduc o a loga i hmic
Hodge decomposi ion, bu an elemen a y p oo was la e ound in [No].
Fo sake o comple eness, we gi e a p oo , which is e en simple han
ha o [No].
Lemma 2.1. [De]Le Xbe a complex p ojec i e mani old, V⊂X
a no mal c ossing hype su ace and ωa global loga i hmic p- o m wi h
pola se con ained in V. Then ωis closed.
P oo : In o de o p o e ha a p- o m is closed, i is sufficien o p o e
ha he es ic ion o he p- o m o a gene ic (p+ 1)-dimensional sub-
mani old is closed. Hence we may assume ha n= dimC(X)=p+1,
and he p oo will be by induc ion on p.
Deg ee o Solu ions o P a Equa ions 597
The case p= 0 is i ial: Ω0
X(log V)=OXand a global holomo phic
unc ion is cons an . Assume now ha he lemma has been p o ed o
(p−1)- o ms.
I ω∈H0(X, Ωp
X(log V)), hen, as in [No], we may conside he
cu en Tωo bideg ee (p, 0) defined by
Tω(φ)=X
ω∧φ,
o e e y smoo h (1,p+ 1)- o m φ. This is well defined, i.e. he in eg al
is con e gen , p ecisely because ωhas loga i hmic poles along V: he
2- o m 1
zdz ∧d¯zis in eg able on he disc. Simila ly, we may associa e o
dω, which is s ill loga i hmic, a cu en Tdω o bideg ee (p+1,0). We
hen ha e, in he sense o cu en s,
∂Tω=Tdω,
which ollows om he ac ha i ψis any smoo h (0,p+ 1)- o m hen
Xd(ω∧ψ) = 0 by S okes heo em (he e ω∧ψis again a cu en , and
he in eg al is i s alue on d1≡0). In pa icula , we ha e
∂Tω≡0⇔dω =0.
On he o he hand, ∂Tωis no ze o: a simple compu a ion, based on
∂(dz
z)=2πiδ0(whe e δ0is he Di ac dis ibu ion), shows ha [No]
∂Tω=2πi TRes(ω),
whe e, wi h a negligible abuse o no a ion, we iden i y TRes(ω)(a cu en
on ˆ
Vo bideg ee (p−1,0)) wi h i s di ec image in X(a cu en o
bideg ee (p, 1)).
By induc ion hypo hesis, Res(ω) is closed, i.e. ∂TRes(ω)≡0, and so
∂∂Tω≡0. By egula i y heo y, he cu en ∂Tωis in ac a holomo phic
(p+ 1)- o m, because i is o bideg ee (p+1,0) and ∂-closed.
By S okes Theo em,
X
∂Tω∧∂Tω=X
d(Tω∧∂Tω)=0
(Tω∧∂Tωis a cu en , being ∂Tωsmoo h, and i s diffe en ial is ∂Tω∧∂Tω
because ∂∂Tω≡0). This o ces ∂Tω o be iden ically ze o, because
∂Tω∧∂Tω≥0.
Rema k. We ha e used he p ojec i i y o Xonly o educe he p oblem
o dimC(X)=p+ 1. In o he wo ds: any loga i hmic p- o m on any
compac complex mani old o dimension p+ 1 is closed.

598 M. B unella, L. Gus a o Mendes
In he con ex o mani olds wi h Pic(X)=Z, we shall use he ollow-
ing well-known ac :
Lemma 2.2. Le Xbe a complex p ojec i e mani old wi h Pic(X)=Z
and V⊂Xa no mal c ossing hype su ace. Le ωbe a global loga i hmic
p- o m wi h poles along Vand Res(ω)≡ 0,1≤p≤n−1. Then Vis
no smoo h.
P oo : By con adic ion, assume ha Vis smoo h, so ha η= Res(ω)≡
0 is a holomo phic (p−1)- o m on V. The line bundle OX(V) is ample,
so ha by Kodai a Vanishing Theo em Hp(X, OX(−V)) = 0, because
p<n. Hence he es ic ion map Hp−1(X, OX)→Hp−1(V,OV)is
su jec i e.
The conjuga e o m ηisa(0,p −1)- o m on Vwhich is ∂-closed,
because ηis ∂-closed. Hence ηdefines a class in Hp−1(V,OV)(`a la Dol-
beaul ), which a ises om Hp−1(X, OX), whence i ollows ha he e
exis s a ∂-closed (0,p−1)- o m βon Xwhose es ic ion o Vis co-
homologous o η, ha is equal o η+∂γ o some (0,p−2)- o m γon
V. A e ex ending γ o Xand eplacing βby β−∂γ, we may and will
suppose ha βcoincides wi h ηon V.
Le now θbeaK¨ahle o m on X. By conside ing ηas a cu en Tη
o bideg ee (p, 1) on X, we may e alua e i on he (n−p, n −1)- o m
θn−p∧β:
Tη(θn−p∧β)=V
η∧θn−p∧β=V
η∧η∧θn−p.
Bu θn−p∧βis ∂-closed and Tη=1
2πi∂Tωis ∂-exac , so ha he in eg al
is ze o. Con adic ion, because η∧η∧θn−p≥0 and η∧η∧θn−p≡ 0.
I will be use ul o e o mula e Lemma 2.1 in a mo e abs ac o m,
due o Bogomolo [Bo], [Re]. To his end, we ecall he defini ion
o Kodai a dimension o a holomo phic line bundle Lo a p ojec i e
a ie y X, deno ed κ(X, L)[Ii]. Conside he ing
R(X, L):=
∞

m=0
H0(X, L⊗m)
and he homogeneous field o ac ions Q(X, L):={li
lj|li,l
j∈H0
(X,L
⊗m),
m≥0}. Then we define κ(X, L) as he anscendence deg ee o Q(X, L),
i R(X, L)=Co κ(X, L):=−∞,i R(X, L)=C. One has κ(X, L)≤
n= dimC(X), and κ(X, L)=ni Lis ample.
Deg ee o Solu ions o P a Equa ions 599
Lemma 2.3. [Bo]Le Xbe a complex p ojec i e n-mani old, V⊂Xa
no mal c ossing hype su ace and L∈Pic(X). I he e exis s a non i ial
global sec ion σo Ωp
X(log V)⊗L, hen
κ(X, L−1)≤p.
P oo : Suppose by con adic ion ha κ(X, L−1)≥p+ 1, i.e. o some
m≥1 he e exis p+ 2 global sec ions l0,...,l
p+1 ∈H0(X, L⊗−m) such
ha he me omo phic unc ions on Xgi en by i:= li
l0a e algeb aically
independen .
Le us suppose o a momen ha m= 1. We can mul iply σby each
li, ob aining global sec ions ωi:= liσ∈H0(Ωp
X(log V)). Since ωi= iω0,
he closedness o each ωi(Lemma 2.1) gi es, o i=1,...,p+1,
d i∧ω0≡0.
The non i iali y o ω0implies d 1∧d 2∧...∧d p+1 ≡0, con adic ing
he algeb aic independence.
The case m>1 is educible o he case m= 1 by passing o a sui able
m- old amified co e ing. We e e o [Re] o de ails.
3. Bounding he deg ee o solu ions
Le σ∈H0(X, Ωp
X⊗N) be a P aff equa ion and V⊂Xa no mal
c ossing hype su ace, which is a solu ion o σ.
We can look a σas a global holomo phic sec ion o Ωp
X⊗OX(−V)⊗
N⊗O
X(V), ha is, a me omo phic sec ion ˆσo Ωp
X⊗O
X(−V)⊗N
wi h simple poles (ˆσ)∞⊂V. Locally, i.e. a e local i ializa ion o
OX(−V)⊗N, we can see ˆσas a me omo phic p- o m ωwi h simple
poles (ω)∞⊂Vand we asse :
Lemma 3.1. The me omo phic p- o m ωis loga i hmic.
P oo : We ha e o check ha ha , i { =0}is a local educed equa ion
o V, hen d ∧ωis holomo phic. F om iVσ≡0 (Defini ion 1.2) i ollows
ha d ∧σis iden ically ze o along V, i.e.
d ∧σ= ·θ,
o some egula local sec ion θo Ωp+1
X⊗N. Hence d ∧σ
is a egula
sec ion o Ωp+1
X⊗Nand d ∧ωis holomo phic, because (up o local
i ializa ion) ω=σ
.
Rema k ha he con e se o Lemma 3.1 is also ue: i ωis a loga-
i hmic p- o m wi h pola se gi en by { =0}, hen V={ =0}is
a solu ion o he P aff equa ion defined by ω. Tha ’s he eason o
600 M. B unella, L. Gus a o Mendes
which he use o loga i hmic o ms is pa icula ly well adap ed o he
s udy o solu ions o P aff equa ions.
The meaning o Lemma 3.1 is ha σis in ac a global holomo -
phic sec ion o Ωp
X(log V)⊗O
X(−V)⊗N. Now, Bogomolo ’s Lemma
(Lemma 2.3) gi es:
κ(X, OX(V)⊗N−1)≤p<dimC(X),(3)
which is he “bound on d(V)” we we e looking o .
Le us now specialize o he case Pic(X)=Z. Then
OX(V)⊗N−1=lH,
whe e l=d(V)−d(N)∈Zand His he posi i e gene a o o Pic(X)=
Z. Since l>0 implies κ(X, lH) = dimC(X), we conclude om (3) ha
d(V)−d(N)≤0
as desi ed. Mo eo e , d(V)−d(N) = 0 means OX(V)⊗N−1=OXand
he e o e he P aff equa ion is globally defined by a (closed!) loga i hmic
p- o m ωwi h (ω)∞=Vand hence Res(ω)≡ 0. In his case, Lemma 2.2
says ha Vis no smoo h.
In o de o apply his esul o olia ions o CPn, we ema k:
Lemma 3.2. Le σ∈H0(CPn,Ωp
CPn⊗N)be a P aff equa ion
associa ed o a olia ion Fo CPnwi h codimCSing(F)≥2(and
codimC(σ)0≥2). Then d(N)=d(F)+p+1.
P oo : Take a gene ic linea subspace Π ≃CPpand conside he es ic-
ion o σ o Π, deno ed iΠσ, wi h hype su ace o ze os on Π deno ed
by (iΠσ)0. Obse e ha iΠσis a global egula sec ion o KCPp⊗N|Π
anishing on (iΠσ)0; hence
OΠ((iΠσ)0)=KCPp⊗N|Π,
whe e KCPpis he canonical line bundle, whose deg ee is −(p+1). Since
(iΠσ)0is he angency se be ween Fand Π and d(F) is defined as he
deg ee o (iΠσ)0in Π, we ob ain d(F)=−(p+1)+d(N).
Re u ning o he gene al inequali y (3), we s ess ha i gi es in-
o ma ions wha e e Pic(X) is. Roughly speaking, i says ha OX(V)
is “pa ially less posi i e” han N. Fi s o all, le us obse e ha i
κ(X, OX(V)⊗N−1)= ≥0 hen (as he p oo o Lemma 2.3 shows) on
a sui able amified co e ing o X he P aff equa ion will be defined by a
global (and closed) loga i hmic p- o m and will ha e algeb aically inde-
penden fi s in eg als. Nex , when Xis a su ace we ha e he ollowing
Deg ee o Solu ions o P a Equa ions 601
ac (p obably, a simila s a emen holds in any dimension). Recall [Dm]
ha a di iso Dis ne i D·C≥0 o e e y i educible cu e C⊂X.
Lemma 3.3. Le Mbe a line bundle on a p ojec i e su ace Xwi h
κ(X, M)≤1. Then he e exis s a non- i ial ne di iso D, wi h eal
coefficien s, such ha M·D≤0.
P oo : Le NSR(X)⊂H2(X, R) be he eal Ne on-Se e i g oup o X,
le Nne ⊂NSR(X) be he ne cone (i.e., he closu e o he ample cone),
and le Npse ⊂NSR(X) be he pseudoeffec i e cone (i.e., he closu e
o he effec i e cone). See o ins ance [Dm] o hese no ions. Then,
by Kleiman c i e ion (in i s dual o m), a line bundle Mbelongs o he
in e io o Npse i and only i M·D>0 o e e y D∈Nne {0}. On he
o he hand, o say ha Mbelongs o he in e io o Npse is he same as
o say ha κ(X, M) = 2, by [Dm, P op. 6.6]. Whence he esul .
In ou case, applied o M=OX(V)⊗N−1 his ac gi es
V·D≤N·D.
On he o he hand, one finds in [B ] he inequali y
V·V≤N·V
(and Vis no necessa ily ne ). The ela ion be ween hese inequal-
i ies is no clea o us: one is a “global” s a emen abou he line
bundle OX(V)⊗N−1o e X, while he o he is a “local” s a emen ,
i.e. abou he es ic ion o OX(V)⊗N−1 o V.
We now discuss some possible ex ensions.
Gi en an analy ic hype su ace V⊂Xwhose singula i ies a e wo s
han no mal c ossings, we can gene alize he defini ion o loga i hmic
o ms (Sec ion 2) in wo ways, which a e no equi alen in gene al:
1) Gi en a me omo phic p- o m ωwi h simple poles along V={ =0},
we say ha ωis weakly loga i hmic i dω (o equi alen ly d ∧ω)is
holomo phic.
2) Gi en ωas in 1), we say ha i is s ongly loga i hmic i , on a
neighbou hood o any x∈V,ωbelongs o he OX-module gene -
a ed by holomo phic o ms and he o ms d 1
1,...,d k
k, whe e V=
{ 1· 2...
k=0}, wi h i educed equa ions o local b anches o V
a x.
The p e ious Lemma a 2.1, 2.3 a e s ill alid o s ongly loga i hmic
o ms, as was obse ed in [No]. The e a e a leas wo ways o see his: