Schlesinger Foliation for Deformations of Foliations

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Schlesinger Foliation for Deformations of Foliations

Yohann Genzmer

To cite this version:

Yohann Genzmer. Schlesinger Foliation for Deformations of Foliations. International Mathematics

Research Notices, Oxford University Press (OUP), 2017, �10.1093/imrn/rnw316�. �hal-01934801�

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FOLIATIONS.

YOHANNGENZMER

Abstrat. Inthisartile,weshowthatfor anydeformationofanalytifo-

liations, there exists a maximalanalyti singular foliationon the spae of

parametersalongtheleavesofwhihthedeformationisintegrable.

ThisworkwaspartiallysupportedbyANR-13-JS01-0002-0.

1. Introdutionandstatements.

ThePainlevéV I êquation^[1℄,^that^motivates^this^work,îsâ^non-linear^dierential

equationofordertwothat parametrizestheisomonodromideformationsoflinear

Fuhsian systemsoverthe four-punturedomplexsphere. This apartiularase

oftheShlesingersystemsthatparametrizestheintegrabledeformationsofGarnier

systems. In both ase, afoliation onthe spae of parametersof thedeformation

whoseleavesparametrizetheintegrabledeformationsisdesribed.

Ourpurposeistohighlightthisphenomenonin ageneralontext.

Theorem. Let(X^m,Fⁿ)−→^π B^p ^be^a^properdeformationofanalytifoliationswith

B^p ^as ^spae^of parameters. Then there exists aunique analyti singular foliation

H ôn B^p ôf ^maximal ^dimension âmong ^those ^that întegrates ^the deformation. In partiular, H ^satises ^the ^following ^property: ^for âny ^leaf L ôf H, ^the ^restrite^d

deformation

(X^m,Fⁿ)|_π−1(L)

−→π L

isintegrable.

Thedenitionofafoliation integrating agivendeformationwill appearbelow.

Ifweremovethemaximality property,then thetheorem beomes trivial. Indeed,

thefoliationofB^p ^by^points^satisesîtsônlusion. ^Moreover,^forâ^generi^defor-

mation, thefoliation H^produed ^by ^the^result ^above ^will ^be ^indeed ^the^foliation

by points. Nevertheless, in viewof the example mentionned in the introdution,

the foliationH^deserves^to ^be^alled ^the ^Shlesinger ^foliation ^of ^thedeformation.

Finally,themaintheoremholdsintherealanalytilassaswellasintheomplex

one.

ItmightbepossiblethatalongsomeexeptionalurvestransversetoH^,^the^defor-

mationisalsointegrable. Suhaurvehastobemoreorlessisolated: theyannot

foliate themanifoldB^p êven^loally. ^Forînstane, ⁱⁿ ^the^framework ôf^the^theory

ofFuhsiansystems,onsiderthesixparametersfamilydenedby

(1.1)

d

dz = A1

z−u1

+ A2

z−u2

+ A3

z−u3

(3)

where Ai =

0 ai

0 0

. Sine the matries Ai ^ommute, ^the ^Shlesinger^system

[9,10℄reduesto

∂Ai

∂uj

= 0, i, j= 1, 2, 3.

Thus,theShlesingerfoliationofthisdeformationisgivenbyda1=^da2=^da3= 0.

However,ifa1=a2= 0^anda36= 0^,^then^(1.1)degeneratestowardasystemwhih isonjugatedto

d

dz =

0 1 0 0

z−u3

.

Therefore,atanypoint(0,0, a3)^witha36= 0^,^the^family^(1.1)îs^stillîntegrableâlong

thewholesubmanifolda1=a2= 0^,^whihîs^notâ^leafôf^the^Shlesinger^foliation.

Hene, in general, the leaf of H ^does ^not parametrize the maximal submanifold alongwhih thedeformationisintegrable.

The main theorem an beompared to the following result of Kiso [4℄: letL ^be

aLie algebraof holomorphivetorelds onaomplexmanifoldN ×M ^tangent

to theprojetiononM^. ÎfL îs^simpleând ôf^nite ^dimension^then^there êxists â

maximalfoliationon N âlong^the^leavesôf^whih Lîs integrable. In ourontext, the Lie algebra underlying the foliation Fⁿ îs ⁱⁿ ^general ^not ôf ^nite ^dimension.

Moreover,the brationπ^is ^not^trivial: ^the^manifold ^supporting^the ^foliation^an

alsobedeformed.

2. Deformationof foliations andKodaira-Spener map.

2.1. Deformations offoliations. LetX^m ^beânânalyti^manifoldândΘX^m îts

tangentsheaf. LetE ^beâôherent^subsheafôfΘX^m. ^The^stakôf Eât âny^point pîs^nitely^generatedâsC^ω(X^m)_p^-module^-^theânalyti^funtionsônB^p ^-ând^we

denote byrankp(E) ^the^minimal ^numberôf êlementsôf â^generating ^family^. ^The

singularlous ofE ^is^dened^by

Sing(E) ={p∈X^m|dim_R{v(p)|v∈Ep}<^rankp(E).}

Sine E îsôherent,^the^singular^lousîs ânânalyti^subsetôf X^m^[7℄. ^The^sheaf

issaidregular ifSing(E) =∅^. ^The^integer d= max

p∈X^mdim_R{v(p)|v∈Ep} ≤m.

isalledthedimensionofE ^andm−d^itsodimension.

Denition1. AnanalytifoliationonX^mîsâôherent^subsheafFôfΘX^m^whih

isintegrable,i.e,

[Fp,Fp]⊂ Fp ^for^anyp∈X^m\^Sing(F).

Denition2. Adeformationoffoliation,denotedby(X^m,Fⁿ)−→^π B^p^,^is^the^data

ofapropermap π:X^m→B^p ^that îsânânalytideformationofmanifold[5℄and ofaregularfoliationFⁿ ^tangent^to ^the^bersôfπ, î.e.,Fⁿ ⊂kerdπ^. ^Theînteger mîs^the^dimensionôfX^m^, p^the^dimensionôfB^p ând n^the^dimensionôfFⁿ^.

ThefollowinglemmaisadiretonsequeneoftheFrobéniustheorem[2℄andstates

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Lemma3. Let (X^m,Fⁿ)−→^π B^p ^be^adeformation of foliation andx∈Xⁿ.^There

existsan isomorphism φ^of deformations offoliation denedonan open neighbor- hoodU ∋x^suh^that ^the ^following ^diagram ^ommutes

(U,Fⁿ|_U) ^φ //

π

%%

❑❑

❑

(R^m−p×B^p, Lⁿ)

pr2

ww

♦♦♦♦♦♦♦♦♦♦♦ π(U)

.

Intheseondtermoftheaboveommutativediagram,thefoliationLⁿ^ofR^m−p×B^p

isgiven bythe bers ofthe projetion

Π :

R^m−p×B^p → R^m−p−n×B^p

(x1,· · · , xm−p, τ) → (x1,· · ·, xm−p−n, τ) .

Figure2.1. Loaltrivializationofadeformationoffoliation.

Denition 4. Let (X^m,Fⁿ) −→^π B^p ^be ^a deformation of foliation. A foliation

H ôf B^p îs ^said ^to întegrate ^the deformationif, for any point τ ∈B^p\^Sing(H)^,

there exists aneighborhood U ∋τ ândâ^regular^foliation G ⁱⁿ π⁻¹(U)^suh ^that dimG=n+ dimHând

Fⁿ|_π−1(U)⊂ G ⊂π^∗ H|_U.

Inpartiular,foranyleafL^ofH,^the^restriteddeformation

π:

π⁻¹(L),Fⁿ|_π−1(L)

→L

is ompletely integrable. In Figure (2.1), the deformation is loally ompletely

integrable: the foliation H ^has ône ^leaf ^that îs H îtself ând ^the ^foliation G ôf

the denition aboveis given in the oordinates of Lemma 3 by the bers of the

projetion

p:

R^m−p×B^p → R^m−p−n

(x1,· · ·, xm−p, τ) → (x1,· · ·, xm−p−n) .

Ourgoalisto showtheexisteneofauniquemaximalintegratingfoliation.

2.2. The sheafof basi vetor elds.

Denition5. Avetoreldv îs^said^to^be^basi^forF îfândônlyîf [v,F]⊂ F.

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Denition 6. Avetoreld v îs^said^to ^be^projetable îfândônlyîf^there êxists

aofvetoreld w^onB^p^suh^that ^the^following^diagram ^ommutes X^m ^π //

v

B^p

w

T X^m ^dπ //T B^p .

Itissaidtobevertialifw= 0,^i.e.,dπ(v) = 0.

WedenotebyB^π^the^sheafôf^basiând^projetable^vetorêldsândB⁰^the^subsheaf

ofbasiandvertialvetorelds. ThesheafB⁰ îsâ^sheafôf^modulesôver^the^ring

ofloalrstintegralsofFⁿ.În^theôordinates^given^by^Lemma^3,â^setionôf^the

quotientsheafB⁰/Fⁿ ^is^written

m−p−n

X

i=1

a(x1, . . . , xm−p−n, τ) ∂

∂xi

.

Hene,

B⁰/Fⁿîsâ^free. Ît^will^be^moreônvenient^to^work^withâC^ω(X^m)^-module,

whihis why, we will onsiderthe produt

B⁰/Fⁿ⊗ C^ω(X^m). Îtîs âlso â^loally

freesheafoverC^ω(X^m)^.

2.3. ThebasiKodaira-Spenermap. Letw^beâ^vetorêldônB^p^denedôn

asmallopensetW ôfB^p^. Âording^to^Lemma^3,^thereêxistsâôvering{Ui}_i∈I ôf

aneighborhoodinXⁿ^ofπ⁻¹(W)^suh^that^thedeformationistrivializedonanyUi

bysomeonjugayφi.^ForânyUi,^the^vetorêldvi=dφ⁻_i¹(0, w)îsâ^setionôfB^π

onUi ^that ^projets^on w.Consideringtheimage of thefamily{vi⊗1−vj⊗1}_ij

inH¹

π⁻¹(W),B⁰/Fⁿ⊗ C^ω(X^m)

,weobtainaC^ω(B^p)−^morphism^of^sheaves

∂Fⁿ: ΘB^p→R¹π∗

B⁰/Fⁿ⊗ C^ω(X^m)

whih is alled the basi Kodaira-Spener map of the deformation. Notie that

this is not thestandard Kodaira-Spener map asdened in [6℄ sine, it does not

measuretheinnitesimaldiretionsoftriviality,but theinnitesimaldiretionsof

integrability.

Lemma7. ker∂Fⁿ îsâ^foliation ôf B^p.

Proof. ThesheafB⁰/Fⁿ⊗ C^ω(X^m)îsâôherent^sheafôfC^ω(X^m)^-modules. ^Sine π îs ^proper, R¹π∗B⁰/Fⁿ ⊗ C^ω(X^m) îs â ôherent ^sheaf ôf C^ω(X^m)_Bp−^modules

, and so is ker∂Fⁿ ^[3℄ ^. ^Suppose w1 ^and w2 ^belongs ^to ^the ^kernel ^of ∂Fⁿ^. ^By

onstrution,thereexisttwofamilies{v^ǫ_i}_i^,ǫ= 1,2^of^setions^ofB^π ^suh^that^for

anyiândǫ^,^the^vetorêldv^ǫ_i ^projetsônwǫ ând^suh^that^forânyi, j ândǫ^, vi^ǫ−vj^ǫ=t^ǫij∈ Fⁿ.

ThediereneoftheirLiebraketsiswritten

v_i¹, v_i²

− v¹_j, v²_j

=

v_j¹, t²_ij

− v²_i, t¹_ij

+ t¹_ij, t²_ij

Sine[B^π,Fⁿ]⊂ Fⁿ ând^sine^the^mapdπômmutes^with^the^Lie^braket,ône^has

∂Fⁿ([w1, w2]) =

[v_i¹, v_i²]⊗1− v_j¹, v_j²

⊗1 _i,j= 0.

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Soker∂Fⁿ îsîntegrableând^thus îsâ^foliation.

3. Integrating foliationsof deformations.

ConsiderthetrivialprojetionR^m−p×R^p→R^p wherethesoureisfoliatedbyLⁿ

givenbythebersof

R^m−p×R^p → R^m−p−n×R^p

(x1,· · ·, xm−p, τ) → (x1,· · · , xm−p−n, τ).

Let{Ti}_i=1,···_,l ^beânînvolutive^familyôf^germsôf^vetorêldsînduingâ^germôf

regularfoliationofdimensionl ⁱⁿ(R^p, τ)^. ^The^family ∂xm−p−n+1,· · ·, ∂xm−p, T1,· · · , Tl

where Ti îs ^seen âs ^vetorêld ⁱⁿ R^m−p ×R^p is involutive and denes an inte- gratingdimensionn+l ^foliationG^. ^Notie ^thatG îs^notûnique^but ^does^depend

only on thebasi part of Ti^: ^forâny^family ôf ^vetorêlds {Xi}_i=1..l ^tangent ^to ∂x_m−p−n+1,· · ·, ∂x_m−p ^,^thedistribution

∂xm−p−n+1,· · · , ∂xm−p, T1+X1,· · · , Tl+Xl

induesthesamefoliationG^. ^This^remarkîs^the^keyôf^the^proofôf^thepropositions below.

Proposition 8. Let (X^m,Fⁿ) −→^π B^p ^be ^a deformation of foliation and w ^be ^a

vetor eldin B^p ^dened ^near τ ^with w(τ)6= 0. ^The^two ^following ^properties ^are

equivalent:

(1) the foliation induedbyw ^integrates^the deformation.

(2) ∂Fⁿ(w) = 0^.

Proof. Suppose that there exists aregular foliation G ^of ^dimensionn+ 1 ⁱⁿ X^m

suhthat Fⁿ|_π−1(U)⊂ G ⊂π^∗H^whereHîs^the^foliationôfB^p îndued^bywⁱⁿ â

neighborhoodU ∋τ^. Âpplying^the^F^robénius^result^toG ^yieldsâôvering{Ui}_i∈I

of π⁻¹(U) ând â ^family ôf ônjugaies {φi}_i∈I ^suh ^that ^the ^following ^diagram

ommutes

Ui,G|_U_i ^φⁱ //

π

%%

❑❑

❑

(R^m−p×B^p,R^m−p× H)

pr2

vv

♠♠♠♠♠♠♠♠♠♠♠♠♠♠

π(Ui)

Byonstrution, thevetoreld vi =dφ⁻¹_i (0, w) îs^basi^for Fⁿ ând ^projetsôn w^. ^Thus ∂Fⁿ(w) ={vi⊗1−vj⊗1}_ij. ^Moreover,vi−vj îs^vertialând ^tangent

toG.^Thus,îtîsâlso^tangent^to Fⁿ.^Hene,∂Fⁿ(w) = 0^.

Now, suppose that ∂Fⁿ(w) = 0^. ^For â ôvering {Ui}_i∈I ôf â neighborhood of

π⁻¹(U)^, ^there êxists â ^family ôf ^projetable ^basi ^vetor êlds {vi}_i∈I ^, vi ∈ ΘXⁿ(Ui) ^suh ^that êah vi ^projets ôn w ând ^suh ^that vi −vj îs ^tangent ^to

Fⁿ^. În^the^loalôordinates^given^by^Lemma^3,^the^vetorêld vi îs^written

vi=

m−p−n

X

i=1

ai(x1,· · ·, xm−p−n, τ) ∂

∂xi

+

m−p

X

i=m−p−n+1

ai(x, τ) ∂

∂xi

+w.

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Sine



vi−

m−p

X

i=m−p−n+1

ai(x, τ) ∂

∂xi

, ∂x_k



= 0

fork=m−p−n+1, . . . , m−p^,^the^family^of^vetor^elds

∂xm−p−n+1,· · ·, ∂xm−p, vi

isinvolutiveanddenesaloal regularintegratingfoliationGi ^of^dimensionn+ 1^.

The foliation Gi ^depends ônly ôn ^the ^basi ^part ôf vi ând vi −vj îs ^tangent ^to

Fⁿ^, ^thus ^the^family {Gi}_i∈I ân^be^glued ⁱⁿ â^global ^foliation^that întegrates^the

deformationoverU^.

Now,weanprovethemainresultofthisartile.

Theorem. Let (X^m,Fⁿ)−→^π B^p ^beâdeformation of foliation. Then, thereexists a unique singular analyti foliation D ôn B^p ôf ^maximal ^dimension âmong ^those

thatintegrate the deformation.

Proof. Let us onsider the sub-sheaf D ôf ΘB^p ^dened ^byv ∈ D îfând ônly ^the

foliationdened by v ^integrates^the deformation. Aording to Proposition 8, D

is a foliation. By onstrution, if D ^has ^the ^property ^of ^the ^theorem, ^it ^will ^be

of maximal dimension for that property. Let d ^be îts ^dimension ând ônsider â

pointpⁱⁿîts^regular^lous. ^The^sheafDîs^loallyâroundp^spanned^byâ^familyôf

exatlyd non-vanishingvetorels D(U) =hw1,· · ·, wdi^, U ∋ p^. ^Now, ^for ^some

overing{Ui}_i∈I ôfâneighborhoodofπ⁻¹(U)^,^thereêxistsâ^familyôf^projetable

basivetorelds {vi,k}i∈I, k=1...d^, vi,k ∈ ΘXⁿ(Ui) ^suh ^that vi,k ^projets^on wk

and vi,k−vj,k îs ^tangent ^to Fⁿ^. În^the ^loal ôordinates ^given^by ^Lemma ^3,^the

familyofvetorelds

∂x_m−p−n+1,· · · , ∂x_m−p, vi,1, . . . , vi,d

isinvolutiveanddenesafamilyofloalintegratingfoliations{Gi}_i∈I ôf^dimension n+d ^that ân^be ^glued. ^Finally, ^weôbtain â^global ^regular integratingfoliation

thatintegratesthedeformationoverU^.

Example9. Linear foliationsonomplex tori ofdimension 2.

Aomplextorusofdimension2^is^given^by^a^lattieΛ^written Λ =Ze1⊕Ze2⊕Ze3⊕Ze4

where

ei=

ei1

ei2

i=1···4

isaR−^free^family^of^four^vetorsⁱⁿC². Theomplex

torusC²/Λ îsêndowed^withâ^linear^foliation^given^by^the^losed1−^form ωα=^dx+α^dy

withα∈P¹.^The^linear^foliationsônômplex^toriôf^dimension²^formâ9−dimensional familyoffoliationswhosespaeofparametersisU ×P¹. Here,U îs^the^real^Zariski

opensetin C⁸givenbytheequation

det

ℜe1 ℜe2 ℜe3 ℜe4

ℑe1 ℑe2 ℑe3 ℑe4

6= 0.

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Anexpliitomputationshowsthat theShlesingerfoliationofthisdeformationis

givenbythelosedsystem

(3.1) H:









 d

e11+e12α e21+e22α

= 0

d_e

11+e12α e31+e32α

= 0

d_e

11+e12α e41+e42α

= 0 .

It is regular on the spae of parameters U ×P¹. This result an also be seen as follows: the representation of monodromy of ωα ôn C²/Λ ômputed ôn ^the

transversallinex= 0^is^written π1 C²/Λ

≃Z⁴→^Aut CP¹

: γi→ y→y+ei1+ei2α

, i= 1,2,3,4

Ahangeofglobaloordinatesy →ay+bôn^thetransversallinex= 0âtsôn^the

monodromythefollowingway

y→y+a(ei1+ei2α) , i= 1,2,3,4

Hene,theonjugaylassoftherepresentationofmonodromyisonstantalonga

deformationifandonlyifthequotients

e11+e12α

e_i1+ei2α, i= 2,3,4 âreônstant,^whihîs

preiselytheondition3.1. Besidesthat,inthisexample,anyleafoftheShlesinger

foliationparametrizesthemaximallousofintegrabilityforanypointin thespae

ofparameters.

ThefoliationHân^beômpatiedâsânâlgebrai^foliationHônCP⁸×CP¹ and thebration

CP⁸×CP¹ → CP⁶ {eij}_i,j, α

→ (e11+e12α, e12, e22, e32, e41, e42)

istransversetoH^exept^over^the^ritial^lousS={(e11+e12α)e42= 0} ⊂CP⁶.

This bration hasompat bers. Thus, aording to theEhresmann's theorem,

thefoliationH^has^the^Painlevé ^property ^with^respet^to ^that^bration^as^dened

in [8℄. Sine, thePainlevéVIhas alsothe sameproperty, itis naturalto address

thefollowingproblem:

Consideranalgebraideformationofanalgebraifoliation. Hasthe

Shlesingerfoliation the Painlevéproperty foran adaptedbration

?

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Y.Genzmer

InstitutdeMathématiquesdeToulouse

UniversitéPaulSabatier

118routedeNarbonne

31062Toulouseedex9,Frane.

yohann.genzmermath.univ-toulouse.fr