Local polar invariants and the Poincaré problem in the dicritical case

(1)

HAL Id: hal-01934813

https://hal.archives-ouvertes.fr/hal-01934813

Submitted on 26 Nov 2018

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Local polar invariants and the Poincaré problem in the dicritical case

Yohann Genzmer, Rogério Mol

To cite this version:

Yohann Genzmer, Rogério Mol. Local polar invariants and the Poincaré problem in the dicritical case.

Journal of the Mathematical Society of Japan, Maruzen Company Ltd, 2018, 70 (4), pp.1419 - 1451.

�10.2969/jmsj/76227622�. �hal-01934813�

(2)

IN THE DICRITICAL CASE

YOHANNGENZMER&ROGÉRIOMOL

Abstrat. Wedevelopastudyon loalpolarinvariantsofplanar omplex

analyti foliationsat (C²,0)^, ^whih ^leads ^to ^the haraterization of seond typefoliationsand ofgeneralizedurvefoliations,as wellasadesription of

the GSV^-index. ^Weâpplyît ^to^the ^Poinaré ^problem ^for^foliations ôn ^the

omplexprojetive planeP²C^, establishing,in the diritialase, onditions fortheexisteneofaboundforthe degreeofaninvariantalgebraiurveS

intermsofthe degreeofthe foliationF^. ^We^haraterize ^theêxisteneôfâ

solutionforthePoinaréproblemintermsofthestrutureofthesetofloal

separatriesofF ôver ^theûrveS^. Ôur^method,ⁱⁿ^partiular,^reovers ^the

knownsolutionforthenon-diritialase,deg(S)≤deg(F) + 2^.

Contents

1. Introdution 1

2. Basinotionsandnotations 4

3. Balanedequationofseparatries 6

4. Polarintersetionandpolarexess 11

5. PolarexessandtheGSV^-index ¹⁵

6. ThePoinaréproblemfordiritialsingularities 18

7. Topologiallyboundedinvariantsofasingularity 21

8. Topologialinvarianeofthealgebraimultipliity 28

Referenes 30

1. Introdution

Let F ^be â ^singular ^holomorphi ^foliation ôn P²C^. ^The ^number ôf ^points ôf

tangeny, withmultipliities ounted,betweenF ^and^anon-invariantline L⊂P²C

isthedegreeofthefoliationandisdenotedbydeg(F)^. ^In^[27℄,^H.^Poinaré^proposed

theproblemofboundingthedegreeofanalgebraiurveS^invariant^byF ⁱⁿ^terms

ofdeg(F)âsâ^stepⁱⁿ ^ndingâ^rational^rstîntegral^forâ^polynomial^dierential

equation in two omplex variables. Known in Foliation Theory as the Poinaré

problem,alongthepastfewdeadesthisproblemhasgainedsomepartialanswers.

In1991, D. Cerveauand A. LinsNeto provedin [13℄ that ifS ^has^at ^most^nodal

1

2010MathematisSubjetClassiation: 32S65.

2

Keywords. Holomorphifoliation,invarianturves,Poinaréproblem,GSV^-index.

3

Work supported by MATH-AmSud Projet CNRS/CAPES/Conyte. First author sup-

ported by a grant ANR-13-JS01-0002-0. Seond author supported by Pronex/FAPERJ and

Universal/CNPq.

(3)

singularitiesthendeg(S)≤deg(F) + 2^,^this^bound^being^reahedîfândônlyîfF îs

alogarithmifoliationoneinduedbyalosedmeromorphi1-formwithsimple

poles. Later, in 1994, M.Carnier obtainedin [10℄ thesame inequality whenthe

singularities of F ôverS âre âll non-diritial, meaningthat thenumber of loal separatries loalirreduibleinvarianturvesisnite. In1997,in theworks

[3℄and[4℄,M.BrunellaformulatedthePoinaréproblemintermsofGSV^-indies,

denedbyX.Gómez-MontJ.SeadeandA.Verjovskyin[19℄asakindofPoinaré-

Hopfindex of therestritionto an invarianturveof avetoreld tangent to F^.

Towit, Brunellashowsthatthebound deg(S)≤deg(F) + 2 ^ours^whenever^the

sum over S ôf ^the GSV^-indies ôf F ^with ^respet ^to ^the ^loal ^branhes ôf S îs

non-negative[4, p. 533℄. It is well known that, in general,the Poinaré problem

hasanegativeanswerinthediritialase(see6.1and6.2below). Someadvanes

in the understanding of the diritial ase have been made in the past years, as

shownin theworks[7,8,16,12,17℄.

Thestudy of globalinvariant urvesleadsus to theuniverse of loal foliations

on (C²,0)^, ⁱⁿ ^whih ^we distinguish two families with relevant properties. First, generalized urve foliations, dened in [5℄ by C. Camaho, A. Lins Neto and P.

Sad, whih are foliations without saddle-nodes in their desingularization. They

havea propertyof minimizationof Milnornumbersand are haraterized, in the

non-diritialase,bythevanishingoftheGSV^-index ^[4,^11℄. ^The^seond^family,

whihontainstherstone,isformedbyseondtypefoliations,introduedbyJ.-F.

MatteiandE. Salemin [22℄, whihmayadmitsaddles-nodeswhen desingularized

providedthattheyarenon-tangentsaddles-nodes,meaningthatnoweakseparatrix

isontainedinthedesingularizationdivisor. Theyareharaterizedbythefatthat

theirdesingularizationsoinidewiththeredutionofthesetofformalseparatries.

These foliations satisfy a property of minimization of the algebrai multipliity

[22,18℄.

Inareentwork[9℄,F.Cano,N.Corralandtheseondauthordevelopedastudy

ofloal polarinvariantsobtaining,in thenon-diritialase, aharaterizationof

generalizedurvesand offoliationsof seond typeaswell asanexpression ofthe

GSV^-indexⁱⁿ^terms^of^theseinvariants. Essentially,thetehniquethereinonsists on alulating the intersetion number between a generi polar urve of a loal

foliationF ând âûrveôf ^formal separatries C^. ^The ^same^number îs ^produed

fortheformalreferenefoliationhavingtheloalequationofCâsâ^rstîntegral.

Thediereneofthese twonumbersis theGSV^-index ^ofF ^with ^respet^toC^. ^In

thisway,theknownanswerstothePoinaréproblemjustmentionedareobtained.

Inthispaperweextendthisapproahtodiritial foliationsthosewithinn-

itelymanyseparatries. Thediultynowliesinhoosinganitesetofseparatries

in order to produesuh areferenefoliation. Thesolutionis to useabalaned

equation of separatries, a onept introdued by the rst author in [18℄ for the

study of the realization problem the existene of foliations with presribed

redutionof singularities and projetiveholonomyrepresentations. Givena loal

foliationFb^at(C²,0)^with^minimal^redution^ofsingularitiesE: (M,D)→(C²,0)^,

an irreduible omponent D ⊂ D ^is ^said ^to ^benon-diritial respetively diritial ifitisinvariantrespetivelynon-invariantbythestrittransform

foliation E^∗Fb^. ^The ^valene ôf D ⊂ D îs ^the ^number v(D) ôf ôther ômponents

(4)

ofDintersetingD^. Â^balaned êquationôfseparatriesturnsoutto beaformal meromorphifuntionFˆ^thatênompasses^theêquationsôfâllîsolatedseparatries theonesrossingnon-diritialomponentsofDâlong^with^theêquationsôf 2−v(D)separatriesassoiatedto eahdiritialomponentD⊂ D^. ^Thisân^be

a negative number, so diritial separatries mayengender poles in the balaned

equation. We anadditionally adjust this denition when a loal set of separa-

tries C ^for Fb îs^xed ⁱⁿ ôrder^to ^get â^balaned êquation âdapted ^to C^. ^This îs

ahievedby rebalaning the numberof diritial separatriesof Fˆ ⁱⁿ ^suh ^a ^way

thatC⊂( ˆF)0^. ^We^develop^these^oneptsⁱⁿ ^setion^3.

In setion 4, our starting point is the extension of the denition of the polar

intersetion number introduedin [9℄toaformalmeromorphi1−^form,^whih^will

normally be dFˆ^, ^where Fˆ îs â ^balaned êquation ôf separatries of the foliation

Fb^. ^Forâ^xed^setôfseparatriesC ôfFb^,^theômparisonôf^the^polarintersetion numbersof Fb ând ^dFˆ^, ^where Fˆ îs â ^balaned êquation ôf separatries suh that

C ⊂ ( ˆF)0^, ^gives ^rise ^to ^the ^polar ^exess ^index, ^denoted ^by ∆p(F, Cb )^. ^This ^non-

negative invariant works as a measure of the existene of saddles-nodes in the

desingularizationofthefoliation: ∆p(F,b (F)0) = 0îfândônlyîfFb îsâgeneralized urve. This isthe ontentof TheoremA, whihextends to thediritial asethe

haraterization ofgeneralizedurvefoliationprovided, in thenon-diritialase,

bythevanishingoftheGSV^-index^with^respet^to^the^omplete^set^ofseparatries.

Atually, insetion 5,TheoremBestablishes alink betweenthepolarexessand

theGSV^-index ^forâônvergent^setôfseparatries:

GSVp(F, C) = ∆b p(F, Cb ) + (C,( ˆF)0\C)p−(C,( ˆF)∞)p,

whereFˆ îsâ^balanedêquationôfseparatriessuhthatC⊂( ˆF)0^. ^Here,(S1, S2)p

standsfor theintersetion numberoftwogermsat pôf ûrvesS1 ând S2 ^dened

by

(S1, S2)p= dimC

Op

(f1, f2)

where f1 ând f2 âre^redued^loal êquationôf S1 ând S2^. ^Notie^that, ^whenFb îs

non-diritialand C îs ^the ômplete ^set ôfseparatries, this givesGSVp(Fb, C) =

∆p(F, Cb )^.

This formulation of the GSV^-index ^enables ^us ⁱⁿ ^Theorem ^C ⁱⁿ ^setion ⁶ ^to

proposeaboundtothePoinaréproblemin termsofloalbalanedsetsofsepara-

tries. ForafoliationFônP²C^havingânînvariantâlgebraiûrveS^,îfd= deg(F)

andd0= deg(S)^,^it^holds d0≤d+ 2 + 1

d0

X

p∈Sing(F)∩S

h

(S,( ˆFp)∞)p−(S,( ˆFp)0\S)p

i ,

where Fˆp îs â ^balaned êquation âdapted ^to ^the ^loal ^branhes ôf S^. ^Besides,

equalityholdsifallsingularitiesof F ôverS âre generalizedurves. Inpartiular, thisinequalityreoversthebound d0≤d+ 2^for^theâses^treatedⁱⁿ ^[13℄ând^[10℄.

Intherightsideofthepreviousformula,theterms(S,( ˆFp)∞)p ^areobstrutions totheexisteneofauniversalbound forthePoinaréproblem. This ispreisely

what happens in two lassialounterexamplesto bedisussed in setion 6: the

(5)

foliationsofdegree1onP²C^given^byω=^d(x^pz^q−p/y^q)^,^withp < q^,^and^the^penil

ofLinsNeto[20℄,afamilyoffoliationsofdegree4admittingrationalrstintegrals

withunboundeddegrees. Intherstase,thetypialberhasaloalbranhat a

diritial singularityrossingadiritial omponentofvalenetwo. Intheseond

family,thegeneriberrepeatedlyrossesradialsingularitiesatanumberoftimes

whihisunboundedwithin thefamily.

Insetion7westudytopologiallyboundedinvariantsofloalfoliationsthose

boundedbyafuntion oftheMilnornumber. Theentralresultofthissetion

Theorem D statesthat aloal urveof separatriesthat ontains,besidesthe

isolatedseparatries,oneseparatrixattahedtoeahdiritialomponentofvalene

oneandaxednumberofseparatriesattahedtodiritialomponentsofvalene

three or higher separatries rossing omponents of valene twoare forbidden

hasaredutionproesswhose lengthistopologiallybounded, thesamebeing

truefor thealgebraimultipliity. This resultis sharp, asshown by theexample

px^dy−qy^dx = 0 ^with p, q ∈ Z⁺ ô-prime. ^Here, ^the ^Milnor ^numberîs ône ând

urvesofseparatrieswithasinglebranhpassingbyaomponentofvalenetwo,

whenq > p >1^,ôr^with^two^branhes^passing^byâ^diritialômponentôf^valene

one,when q > p= 1 ^may^be^obtained^with^redution^trees^of arbitrarilylarge length. Returningto thePoinaré problem, in Theorem E we use the inequality

ofTheoremCinordertoprovetheexisteneofaboundforthePoinaréproblem

whenevertheloalbranhesofthealgebraiurveS ^are^subjet^to ^the^onditions

of topologial boundedness of Theorem C. This result espeially indiates that

the two lassial ounterexamples for the Poinaré problem just mentioned oer

essentially the two waysto violate the existeneof abound: either by means of

highly degenerated separatries rossing diritial omponents of valenes one or

two, or through a multiple branhed urve of separatries attahed to diritial

omponentsofothervalenes.

We lose this artile with setion 8, where we apply loal polar invariants on

a result on the topologial invariane of the algebrai multipliity of a foliation.

InTheoremFweprovethat,forloalfoliationsat(C²,0) ^having^only^onvergent

separatries,the propertyof beingseond lass andthe algebraimultipliity are

topologial invariants. This extends similar results in [5℄, for generalized urve

foliations,and in[22℄,fornon-diritialseondlassfoliations.

2. Basinotionsand notations

A germ of formal foliation Fb ⁱⁿ C² îs ^theôbjet^dened ^by â^germôf ^formal 1−^formât 0∈C²

ˆ

ω= ˆa(x, y)^dx+ ˆb(x, y)^dy,

wherea,ˆ ˆb∈C[[x, y]]^. Â ^sepâratrix ^forFb îsâ^germôf^formalîrreduibleînvariant

urve. If S îs ^dened ^by â ^formal êquation fˆ^, ^then ^the învariane ôndition îs

expressedalgebraiallyas

fˆ^dividesωˆ∧^dfˆⁱⁿC[[x, y]].

A formalfoliationissaidto be non-diritial when ithasnitely manyseparatri-

es. From now on, E : (M,D) → (C²,0) ^stands ^for ^the ^proess ^of ^redution ^of

(6)

singularities ordesingularization of Fb ^(see ^[29℄ ândâlso ^[5℄), ôbtained^by â^nite

sequene of puntual blowing-ups having D= E⁻¹(0) ^as^the ^exeptional^divisor,

formedbyaniteunionofprojetivelineswithnormalrossings. Inthis proess,

all separatriesof Fb ^beome ^smooth, ^disjoint ^and ^transverse ^to D^, ^none ^of ^them

passingthoughaorner. Besides,thesingularitiesalongDôf^the^pull-bak^foliation E^∗Fb ^beome^redued ôr^simple,^whih^means^that,ûnder^some^loal^formal^hange

ofoordinates,theirlinearpartsbelongtothefollowinglist:

(i) y^dx−λx^dy^, λ6∈Q⁺^;

(ii) x^dy^.

Intherstasewhihisreferredtoasnon-degenerate,therearetwoformal

separatriestangentto theaxes{x= 0} ând{y= 0}^. În^theônvergentâtegory^,

bothseparatriesonverge. Intheseondase,thesingularityissaidtobeasaddle-

node. Theformalnormalformforsuhasingularityisgivenin[22℄: uptoaformal

hangeof oordinates,thesingularityisgivenbya1^-form ^of^the^type

(2.1) ζx^k−k

y^dx+x^k+1^dy, k∈N^∗, ζ ∈C.

The invariant urve {x = 0} îs âlled ^strong învariant ûrve ^while {y = 0} îs

theweak one. Inthe onvergentategory, thestrong separatrixalwaysonverges

whereas theweak separatrixmay bepurely formal, as in the famous example of

Euler

(x−y)^dx+x²^dy,

wherey(x) =P

n≥1(n−1)!xⁿ îs^the^Tâylorêxpansionôf^the^weakînvariantûrve.

Theintegerk+ 1>1 ^will^beâlled^weak îndex ôf^thesaddle-node.

If, in theredution proess, the germof thedivisor D ^happens ^to ^ontain^the

weak invariant urve of some saddle-node, then the singularity is said to be a

tangent saddle-node. A non-tangent saddle-node is also said to be well-oriented

withrespetto D^. ^We^will ^denote ^byT(Fb)^the^set ^of ^all^tangentsaddle-nodesof

Fb^. ^Fôllowing^[22℄,^we^say^that^the^foliationFbîsⁱⁿ ^the^seond^lass ôrîsôf^seond

type whennoneofthesingularities ofE^∗Fb ^overD^are^tangentsaddle-nodes.

LetFb ^beâ^germôf^foliation^having(S, p)âsâ^germôf^formal^smoothînvariant

urve. Take loal oordinates (x, y) ⁱⁿ ^whih S îs ^the ûrve {y = 0} ând p ^the

point(0,0)^. ^Letω= â(x, y)^dx+ ˆb(x, y)^dy ^beâ^dening1^-form^forFb^. ^Theînteger

ord0ˆb(x,0)îsâlled^the^tangenyîndex ôfFbâtp^with^respet^toSândîs^denoted^by

Ind(F, S, p)b ^. ^Thisîsânînvariantâssoiated^toFbândS^,independentofthehoies made. IfSîs^weak^separatrixôfâsaddle-node,thenInd(Fb, S, p)>1îs^preisely^the

weakindex. Ontheotherhand,ifSîsêither^the^strong^separatrixôfâsaddle-node oraseparatrixofanon-degeneratereduedsingularity,thenInd(F, S, p) = 1b ^.

In theexeptionaldivisor D^, ^we ^denote ^by^Di(F)b ^the ^set ^of ^diritial ^ompo-

nents, omprisingallprojetivelines generiallytransverseto E^∗Fb^. Â ^separatrix S ôf Fb îs ^said^to ^beîsolated îfîts^strit ^transformE^∗S ^does^not^meet â^diritial

omponent. This onept is well dened as long as we x a minimal redution

of singularities for Fb ^see ^that, ⁱⁿ ^the ^denition ^of ^redution ^of singularities, therearealsoonditionsonthedesingularizationoftheseparatries. Wedenoteby

I(F)b ^the^set^of^isolatedseparatries. Ontheotherhand,aseparatrixwhosestrit

(7)

assoiated to D ∈ ^Di(F)b ^is ^denoted ^by ^Curv(D)^. ^Finally, v(D) ^stands ^for ^the

valene ofD∈^Di(F)b ^,^denedâs^the^numberôfômponentsôfDintersetingD^.

Remark2.1. Theabovedenitionsanbeformulatedintheonvergentategory

andit mayseemsomewhatstrangetointroduethem in theformalategory. In-

deed,foronvergentfoliations,thenaturalandgeometrinotionofleavesdoesexist

whereasonlythenotionofseparatrixmakessensein theformalsetting. However,

the interestand the need to work with formal objetswill beevident assoon as

theoneptofbalanedequation isdened.

3. Balaned equationof separatries

With aslighthangeinthedenition,wefollow[18℄intheoneptbelow.

Denition3.1. Abalanedequation ofseparatries foragermofformalsingular

foliationin (C²,0)^is^a^formal^meromorphi^funtion Fˆ ^whose^divisor^has^the^form ( ˆF)0−( ˆF)∞ = X

C∈I(F)b

(C) + X

D∈^Di(F)b

X

C∈^Curv(D)

aD,C(C),

where,foreverydiritialomponentD⊂ D^,^the^oeientsaD,C ∈ {−1,0,1}^are

zeroexeptfornitely manyC∈^Curv(D)^and^satisfy^the^following^equality:

(3.1)

X

C∈^Curv(D)

aD,C = 2−v(D).

ThebalanedequationofseparatriesFˆ îs^said^to^beâdapted ^toâûrveôf^separa-

triesC ^ifC⊂( ˆF)0^.

Sine aD,C ^belongs ^to {−1,0,1}^, ^the ^funtion Fˆ ^has ^redued ^zeros ^and ^poles

withoutmultipliities. Forinstane,theradialfoliationgivenbyx^dy−y^dx^has^only

onediritialomponentwhosevaleneis0.^ThusF =xyîsâ^balanedêquationôf

separatries. However,F =xy(x−y)/(x+y)îsâlsoâ^balanedêquation,âdapted

to theurveC={x−y = 0}^. ^If Fb ^is non-diritial,then abalanedequation is nothingbutanyequationoftheniteset ofseparatries.

Wereallsomebasifats aboutbalanedequationsof separatriesestablished

in[18℄. Firstadenition:

Denition 3.2. LetFb ^beâ^formal ^foliationât(C²,0) ândE : (M,D)→(C²,0)

beaminimalproessofredutionofsingularities. Thetangenyexess ofFb^along D^is^the^number

(3.2) τ(F) =b X

q∈T(E^∗F)b

X

D∈V(q)

ρ(D)(^Ind(E^∗F, D, q)b −1),

whereq^runsôverâll^the^tangentsaddle-nodesofE^∗FbândV(q)^stands^for^the^set

of irreduible omponents of D^ontaining ^the^pointq^. ^The^numberρ(D)^stands

forthemultipliity ofD^,^whih^oinides^with^the^algebraimultipliityofaurve

γ ât(C²,0)^suh^thatE^∗γîstransversalto DôutsideâôrnerôfD^.