Chern–Moser
operators
and
polynomial
models
in CR geometry
Martin Kolara,∗,1,Francine Meylanb,2, Dmitri Zaitsevc,3 a DepartmentofMathematicsandStatistics,MasarykUniversity,Kotlarska2, 611 37Brno,CzechRepublic
bDepartmentofMathematics,UniversityofFribourg,CH1700Perolles, Fribourg,Switzerland
cSchoolofMathematics,TrinityCollegeDublin,Dublin2,Ireland
Weconsiderthefundamentalinvariantofarealhypersurface
inCN –itsholomorphicsymmetry group –andanalyze its
structureatapointofdegenerateLeviform.Generalizingthe
Chern–Moser operator to hypersurfaces of finite multitype,
we compute the Lie algebra of infinitesimal symmetries of
the modeland provide explicit description for each graded
component. Compared witha hyperquadric,it may contain
additional components consisting of nonlinear vector fields
definedintermsofcomplextangentialvariables.
Asaconsequence,weobtainexactresultsonjetdetermination
for hypersurfaces with such models. The results generalize
directly the fundamental result of Chern and Moser from
quadraticmodelstopolynomialsofhigherdegree.
* Correspondingauthor.
E-mailaddresses:[email protected](M. Kolar),[email protected](F. Meylan),
[email protected](D. Zaitsev).
1 ThefirstauthorwassupportedbytheProject CZ.1.07/2.3.00/20.0003oftheOperationalProgramme
EducationforCompetitivenessoftheMinistryofEducation,YouthandSportsoftheCzechRepublic.
2 ThesecondauthorwassupportedbySwissNationalScienceFoundation Grant2100-063464.00/1. 3 ThethirdauthorwassupportedinpartbytheScienceFoundationIrelandGrant 10/RFP/MTH2878. Published in $GYDQFHVLQ0DWKHPDWLFV ±
which should be cited to refer to this work.
1. Introduction
The holomorphic symmetry group of the unit sphere in C2 has been known since
the seminal work of Poincaré [26]. For general signature (and dimension), computing thesymmetrygroupofareal hyperquadricinCN isthefundamentalstartingpointfor
thestudy ofCRinvariantsofLevinondegeneratehypersurfaces[2,7,10,14,18,25,29–32]. Our aim inthis paper isto analyzesymmetry groupsfor polynomialmodels of higher degree.
Hypersurfaceswith higherdegreemodels arenecessarilyLevidegenerate. Thestudy ofsuchmanifoldshasbeeninitiatedbytheworkofJ.J. Kohninthecontextofboundary regularityofthe ∂ operator, ¯ andhas leadtomajor advancesinanalysis and geometry, forexampleintroducingmultiplieridealsheaves[20,21]andY.-T. Siu’scelebratedworks oninvarianceofplurigenera[27].
Local CRgeometry of Levi degeneratehypersurfaces presents completely new chal-lenges, which are often closer to algebraic, rather than to differential geometry. In particular,iftheLeviformchangesranknearthegivenpoint,thedifferentialgeometric approachofCartan,ChernandTanaka isnotavailable.
The Chern–Moser operator (as defined in[10]) turned out to be themost powerful algebraictool forunderstandinglocalCRgeometryat aLevinondegeneratepoint.The Chern–Moser normalformconstructionessentiallyreduces totheanalysis ofthekernel andtheimageofthisoperator.Ithasbeenalongopenquestionwhethersuchtechniques can be generalizedalso to theLevi degeneratecase[1,3,11–13,15,17,31]. Letus remark thatthe caseof CR manifolds ofhigher codimension has been also intensively studied (seee.g.[4,5,16,19]).
In complexdimension two,acomplete normalform forhypersurfaces offinite type, based on ageneralization of the Chern–Moser operator, was given by the first author in[22].InthepresentpaperweshowthattheChern–Moseroperatorcanbegeneralized inanaturalway toawideclassofLevidegeneratemanifoldsinCN,namelythe
hyper-surfacesoffiniteCatlinmultitype.Weanalyzethekernelofthisoperator,whichcarries completeinformationabouttheinfinitesimalautomorphismsofthemodelhypersurface, andasaconsequencegivessharpresultsonjetdeterminationfortheautomorphismsof thehypersurfaceitself.
LetusrecallthatmultitypeisanessentialCRinvariantwhichCatlindefinedandused toprovesubellipticestimatesonpseudoconvexdomains(hispapers[8,9]).Inparticular, if a subelliptic estimate holds on apseudoconvex model, then it is of finite multitype andholomorphicallynondegenerate, andourresultscan beapplied.Ontheotherhand, we makeno pseudoconvexityassumptions (multitype wasextendedto thegeneralcase in[23]),similarly astheworkofChernandMoserconsiders modelhyperquadricsofall signatures.
Since finite multitype formalizes both the notion of model and invariantly defined weights, both essential for Chern–Moser theory, it provides a natural setting for its extensiontothedegeneratecase.NotethathypersurfacesoffiniteCatlinmultitypemay
containcomplexvarieties,providingapotentiallinkbetweeninvariantsofsuchvarieties and CRinvariantsof thecorrespondinghypersurface.
Our first result deals with a hypersurfacegiven by ahomogeneous polynomial. Let Cν[z] denotethespace ofholomorphic homogeneouspolynomials inz = (z1, . . . , zn) of
degree ν. Recallthatthesharpconditiongeneralizing Levi-nondegeneracyforthe auto-morphism groupbeing finite-dimensionalisthe holomorphic nondegeneracy introduced
by N. Stanton.A real-analytic hypersurfaceM is bydefinitionholomorphically nonde-generate ifnopoint of M admits aholomorphicvectorfieldinitsneighborhood,whose both realandimaginarypartsaretangentto M .
Theorem 1.1. Let P (z, z) be a ¯ homogeneous polynomial without pluriharmonic terms of degree d ≥ 2, such that the hypersurface
MP :=
Im w = P (z, ¯z), (z, w)∈ Cn× C, (1.1)
is holomorphically nondegenerate. Then the Lie algebra g of all germs of infinitesimal automorphisms of MP at 0 admits the weighted grading
g = g−1⊕ g−1/d⊕ g0⊕
d−2
τ =1
gτ /d⊕ g1−1/d⊕ g1, (1.2)
and we have the following explicit description of the graded components:
(1) g−1={a∂w: a ∈ R}, (2) g−1/d={jaj∂ zj + g(z)∂w: aj∈ C, g∈ Cd−1[z], 2i(ajPzj+ ¯ajPz¯j)= g− ¯g}, (3) g0={jfj(z)∂zj+ aw∂w: fj ∈ C1[z], a ∈ R, (fjPzj + ¯fjPz¯j)= aP}, (4) dτ =1−2gτ /d = d−2 τ =1{ jfj(z)∂zj: fj ∈ Cτ +1[z], (fjPzj + ¯fjP¯zj)= 0}, (5) g1−1/d = { j(fj(z)+ ajw)∂zj + g(z)w∂w: aj ∈ C, fj ∈ Cd[z], g ∈ Cd−1[z], jaj∂zj+ g(z)∂w∈ g−1/d, (fjPzj+ ¯fjP¯zj+ 2iP (ajPzj+ ¯ajPz¯j))= 2iP (g + ¯g)}, (6) g1={ jfj(z)w∂zj + aw2∂w: fj ∈ C1[z], a ∈ R, jfj(z)Pzj = aP}.
Note thatpossible pluriharmonicterms in the expansionof P can always be easily eliminated bysimplebiholomorphic changeofcoordinates.Moredetaileddescriptionof the individualcomponents is given in Sections 4, 5 and 6. In the Levi nondegenerate case the correspondingdecomposition contains only fivecomponents, since d = 2 (see
Examples 3.8 and 5.6formanifoldswhichadmitautomorphisms oftheform(4)). Calculationsshowthat thecomponentg1 isalwaysatmost1-dimensional,infactthe
polynomials fj areuniquely determined bya from theequation in (6).Manifolds with nontrivialg1 arecharacterized inTheorem 4.7.
Asaconsequence,weobtainaprecisedescriptionofthederivativesneededto charac-terizeanautomorphism ofageneralhypersurfacewhosemodelisof theform(1.1).Let
M be givennear p by
Im w = P (z, ¯z) + o|z|d,Re w, (1.3)
where P is ahomogeneouspolynomialwithoutpluriharmonictermsofdegree d ≥ 2.We willdenoteby(f1, f2, . . . , fn, g) the componentsofanautomorphismof M (as in(2.13)).
Theorem1.2. The automorphisms of M at p are uniquely determined by
(1) the complex tangential derivatives ∂∂z|α|αfj up to order d − 1,
(2) the first and second order normal derivatives ∂fj
∂w for j = 1, . . . , n, ∂g ∂w,
∂2g ∂w2.
Thisjetdeterminationresulthereis sharp,asshownbyExample 3.8.
Next we consider the more generalcase of a weighted homogeneous model of finite Catlin multitype. Let p ∈ M be a point of finite Catlin multitype (m1, . . . , mn) (see
Section2).As showninSection2, onecan find coordinates(z1, . . . , zn, w) with weight
of zj equalto μj = m1j,weightof w equal to1,suchthat M is locallygivenby
Im w = P (z, ¯z) + F (z, ¯z, Re w), (1.4) where P is aweighted homogeneouspolynomialofweighted degree1 and F has Taylor expansionwithtermsofweighteddegree > 1. Togivethesimplestexamplewithunequal weights,considertheholomorphicallynondegeneratehypersurfaceinC3 definedby
Im w = |z1|2+|z2|4, (1.5)
wheretheweightsare μ1= 12, μ2= 14(afiniteexplicitalgorithmforcomputingmultitype
isgiven in[23]).Note thatwith thechoiceofweights μ1= μ2= 12,the modelbecomes
holomorphicallydegenerate.
Fortherestofthissection,assumethat M is givenby(1.4),andtheassociatedmodel hypersurface
MP :=
Im w = P (z, ¯z) (1.6) isholomorphicallynondegenerate.Let E denote theset
E = n j=1 kjμj; kj ∈ N ∪ {−1} ∩ (0, 1). (1.7)
Theorem 1.3. The Lie algebra of infinitesimal automorphisms g = aut(MP, 0) of MP
admits the weighted grading given by
g = g−1⊕ n j=1 g−μj⊕ g0⊕ η∈E gη⊕ g1. (1.8)
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Wehavea completely analogousexplicit descriptionofthegraded componentsasin
Theorem 1.1.Note thatthefourthcomponentin(1.8)correspondstoparts(4)and(5) of Theorem 1.1(ingeneralitcannotbe splitinthisway,as Example 6.4shows).
Asaconsequence,weobtainthefollowingtheoremthatgivesasharpcharacterization of theautomorphisms of M .
Theorem 1.4. The automorphisms of M at p are uniquely determined by their jets of weighted order 2.
A precise statement giving exactly which derivatives are needed to determine an automorphism isgiveninTheorem 7.1.FortheLevinondegenerate case,when μj =12,
j = 1, . . . , n, we recover exactly the sharp statement of Chern and Moser contained in[10](Corollary 7.2).
Let us remark that most of the results of Sections 4, 5, 6 apply in a more general case, foran arbitraryhypersurface withaweighted homogeneousmodelwhich is holo-morphically nondegenerate(theweights neednotcoincidewith themultitypeweights). However, thefundamentalproperty of theChern–Moser operator(2.16),providing the leading linear part of the transformation law, fails in this case. Hence the infinitesi-mal automorphisms ofthe modelhypersurfacenolonger controlautomorphismsof the hypersurface itself, and there exist examples for which the conclusion of Theorem 1.3
fails.
Thepaper isorganizedas follows.InSection2, werecallthenotionof Catlin multi-typeof asmooth hypersurface M ⊂ Cn+1. WealsostudythegeneralizedChern–Moser
operator, and show how to reduce the weighted jet determination problem for the automorphism group of M , to the study of the set of real-analytic infinitesimal CR automorphisms of MP at p (Proposition 2.15). In Section 3, we introduce the notion
of rigid vector fields and prove results regarding the determination problem for such infinitesimal automorphisms (Theorem 3.3andLemma 3.4).InSections4,5,and 6,we study theinfinitesimalautomorphisms whicharenotrigid (Theorem 4.7,Theorem 5.5, and Theorem 6.2).InSection7,wecomplete theproofsof themain results.
2. TheCatlinmultitypeandgeneralizedChern–Moseroperators
Inthis sectionwerecallthenotionofCatlinmultitypeandconsider ageneralization of theChern–Moser operatoronLevidegeneratehypersurfacesoffinitemultitype.
Let M ⊆ Cn+1beasmoothhypersurface,and p ∈ M beapointof finite type m in the
sense ofKohn and Bloom–Graham[6]. Wewill consider local holomorphic coordinates (z, w) vanishing at p, where z = (z1, z2, . . . , zn) and zj = xj + iyj, w = u + iv. The
hyperplane{v = 0} isassumedtobetangentto M at p, hence M is describednear p as
thegraphofauniquelydetermined realvaluedfunction
v = ψ(z1, . . . , zn, ¯z1, . . . , ¯zn, u), dψ(p) = 0. (2.1)
Usingaresultof[6], wemayassumethat
ψ(z1, . . . , zn, ¯z1, . . . , ¯zn, u) = Pm(z, ¯z) + o
|u| + |z|m, (2.2)
where Pm(z,z) is ¯ anonzerohomogeneouspolynomialofdegree m with nopluriharmonic
terms.
The definition of multitype involves rational weights associated to the variables
w, z1, . . . , zn inthefollowing way.
The variables w, u and v are given weight one, reflecting our choice of variables given by (2.1). The complex tangential variables (z1, . . . , zn) are treated according to
thefollowingdefinitions(formoredetails,see[23]).
Definition2.1.Aweightisan n-tuple ofnonnegativerationalnumbers Λ = (λ1, . . . , λn),
where0≤ λj≤ 12,and λj≥ λj+1.
Let Λ = (λ1, . . . , λn) be aweight,and α = (α1, . . . , αn), β = (β1, . . . , βn) be
multi-indices. Theweighted degree κ of amonomial
q(z, ¯z, u) = cαβlzαz¯βul, l∈ N, isdefinedas κ := l + n i=1 (αi+ βi)λi.
A polynomial Q(z, z, u) is ¯ Λ-homogeneous of weighted degree κ if it is a sum of monomialsofweighted degree κ.
Foraweight Λ, theweighted lengthofamultiindex α = (α1, . . . , αn) isdefinedby
|α|Λ:= λ1α1+· · · + λnαn.
Similarly,if α = (α1, . . . , αn) andα = (ˆˆ α1, . . . , αˆn) aretwomultiindices,theweighted
lengthofthepair(α,α) isˆ (α, ˆα)
Λ:= λ1(α1+ ˆα1) +· · · + λn(αn+ ˆαn).
Theweightedorder κ ofadifferentialoperator
D = ∂|α|+|ˆ α|+l ∂zα∂ ¯zαˆ∂ul isequalto κ := l + (α, ˆα) Λ.
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Definition 2.2. A weight Λ will be called distinguished for M if there exist local holo-morphic coordinates(z, w) in whichthedefiningequationof M takes form
v = P (z, ¯z) + oΛ(1), (2.3)
where P (z, z) is ¯ anonzero Λ – homogeneouspolynomial of weighted degree1 without pluriharmonicterms,and oΛ(1) denotesasmoothfunctionwhosederivativesofweighted
order lessthanor equalto onevanish.
Thefactthatdistinguishedweightsdoexist followsfrom(2.2).Forthese coordinates (z, w), wehave Λ = 1 m, . . . , 1 m .
Inthefollowingweshallconsiderthelexicographicorderonthesetof n-tuples defined as follows: (α1, . . . , αn) < (β1, . . . , βn) wheneverforsome1≤ k ≤ n, αj = βj for j < k
but αk< βk.
Werecallthefollowingdefinition,due toD. Catlin[8].
Definition2.3.Let ΛM = (μ1, . . . , μn) betheinfimumofallpossibledistinguishedweights
Λ with respectto thelexicographicorder.Themultitypeof M at p is definedtobe the
n-tuple (m1, m2, . . . , mn), where mj = 1 μj if μj = 0, ∞ if μj= 0.
Furthermore, ifnoneofthe mj isinfinity,wesaythat M is of finite multitype at p.
Since thedefinitionof multitype includesalldistinguishedweights,theinfimum isa
biholomorphic invariant.
Definition2.4.Coordinatescorrespondingtothemultitypeweight ΛM,inwhichthelocal
description of M has form(2.3), with P being ΛM-homogeneous, are called multitype
coordinates.
Notice thatif M is of finite multitype at p, the infimum is attained, which implies thatmultitypecoordinatesdoexist[8,23].
If M ⊂ C2,then M is offinite typeat p if and onlyif M is of finitemultitype at p.
Inthiscase,thetypeof M at p is equaltothemultitypeof M at p.
From nowon,weassumethat p ∈ M givenby(2.3)isapointof finite multitype
(m1, m2, . . . , mn),
where mj= μ1j, thatis,
v = ψ(z, ¯z, u) = P (z, ¯z) + oΛM(1). (2.4)
Werecallthefollowingdefinitiongivenin[23].
Definition2.5.Let M be givenby(2.4).Wedefineamodelhypersurface MP associated
to M at p by
MP =
(z, w)∈ Cn+1 v = P (z, ¯z). (2.5) Notethatmultitypecoordinates(z, w) are notunique.Neverthelessitisshownin[23]
that all models are biholomorphically equivalent (in fact by apolynomial transforma-tion).
Thefollowingpropositiongivesausefulpartial normalizationof P (cf.[23]).
Proposition2.6. Let ΛM be as in Definition 2.3and P as in(2.3). Then after a polynomial
change of coordinates preserving the weights, we can assume that for every 1≤ k ≤ n,
the following hold:
(1) the derivatives of P satisfy
Pzk|zk+1=···=zn=0 = 0; (2.6)
(2) the expansion of P contains a nontrivial monomial czγkz¯γkwith γkk ≥ 1, γjk =γjk = 0
for j > k, and no other monomial of the form ezγ1k 1 . . . z γkk−1 k−1 z γkk−1 k z αk+1 k+1 . . . z αn n z¯γ k . (2.7)
Proof. Forreader’sconvenience,wefirstprovethestatement(1)for k = 1. Sincethetype is m, wemusthave μ1= 1/m.Choose k ≥ 1 tobethelargest l such that μ1=· · · = μl.
Weclaimthat
P|zk+1=···=zn=0 = 0. (2.8)
Indeed, since the type is m, there exists a nontrivial monomial of degree m in the right-hand side of (2.3). The latter cannot be in oΛ(1) because all μj ≤ 1/m. Hence
that monomial appears inthe expansion of P . Furthermore, by our choice of k, this monomial cannotcontain zj with j > k,because otherwise, itsweighted degreewould
belessthan1.Thisprovestheclaim(2.8).Then,agenericlinearchangeofthevariables
z1, . . . , zk preservestheweight Λ and achieves(2.6)for k = 1.
Now weprovethe statement(1)for generalk = k0.As before,choose k ≥ k0 to be
thelargest l≥ k0 suchthat μk0 =· · · = μl.Withthat k we claimthat
Pzl|zk+1=···=zn=0 = 0 for some k0≤ l ≤ k. (2.9)
Indeed,otherwise P|zk+1=···=zn=0 dependsonlyon z1, . . . , zk0−1.Thenwe candecrease
the weightμk and possiblyincreasethe weights μj for j > k, so thatthenew weight
Λ becomessmallerinthelexicographicorder.Thiscontradicts thechoiceof Λ in Def-inition 2.6 andhenceprovesthe claim(2.9).Then,again agenericlinearchangeof the variables zk0, . . . , zk preserves theweight Λ and achieves(2.6)for k = k0.
To show (2), in view of (2.6) there exists nontrivial monomial czγkz¯γk with γk k ≥
1, γk
j = γkj = 0 for j > k, in the expansion of P . Among all such monomials, we
consideroneswithlexicographicallymaximalγk,andthenamongthosetheone(uniquely
determined) with lexicographicallymaximal γk,which wedenote by czγkz¯γk
. Consider apolynomialweighted homogeneous transformation
zj= zj for j = k, zk = zk+ Cαz αk+1 k+1 . . . z αn n . (2.10) Thenexpanding c(z)γk(¯z)γk
weobtainmonomialsoftheform(2.7)with e = cγk kCα.It
nowsufficestoshowthatnoothertermscancontributetothesamemonomials.Indeed, all other terms bzβz¯β in the expansion of c(z)γk(¯z)γk
will either have smaller β or
the same β but smallerβ. Moreover, by our choiceof the monomial czγk¯zγk, allsuch monomials with γkk ≥ 1, γkj = γjk = 0 for j > k, will also have in the expansion of
P (z, ¯z) terms bzβz¯β witheithersmaller β or thesameβ but smaller β. Finally, ifwe
expandanothermonomial a(z)δ(¯z)δin P (z, z¯),theneither(1) δ
j ≥ 1 forsome j > k,
inwhichcasewecannotgetaterm(2.7),or(2)δj = 0 forall j > k and δj≥ 1 forsome
j > k. Inthesecondcase,inordertohaveaterm(2.7)intheexpansion,we musthave
δj = γjk for j < k and δk≥ γkk,andδ = γk.Butthen a(z)δ(¯z)δwouldbeofweight > 1
contradicting thechoiceof P . 2
Wenowdefine thenotionof weighted jets.
Definition 2.7. Let(z,w) ∈ Cn+1 bemultitype coordinatesand let F :Cn+1→ C bea holomorphicfunctiongiveninthesecoordinates.Theweightedjetof F at p of weighted order κ is given bythefollowing set
∂|α|+|β|F
∂zα∂wβ (p),|α|ΛM+|β| ≤ κ
. (2.11)
Definition 2.8. Let F1, F2 : Cn+1 → C be two holomorphic functions given in some
multitype coordinates.Wesaythat F1 and F2 areweightedequivalentmodulo κ at p if
∂|α|+|β|F1
∂zα∂wβ (p) =
∂|α|+|β|F2
∂zα∂wβ (p), |α|ΛM+|β| ≤ κ.
Wehavethefollowing lemma.
Lemma 2.9. The notion of weighted equivalence modulo κ at p is independent of the choice of multitype coordinates.
Proof. This is adirect applicationof Theorem 4.1 of [23] combined with theLeibnitz rule.Indeed,Theorem 4.1saysthatanybiholomorphictransformationtakingmultitype coordinates(z, w) into multitype coordinates(z, w) hastobeof thefollowing form
zj= zj+ |α|ΛM=μj Cαzα+ oΛM(μj), w= w + c |α|ΛM=1 Dαzα+ oΛM(1), (2.12)
for c ∈ R\ {0},where oΛM(μj) denotestermsintheTaylorexpansionofweighteddegree
greaterthan μj. 2
WewillnowintroducethenotionofgeneralizedChern–Moser operator.
Denote by Aut(M, p), the stability group of M , that is, those germs at p of biholo-morphisms mapping M into itself and fixing p, and by aut(M,p), the set of germs of holomorphicvectorfields inCn+1whosereal partistangentto M .
If M admits aholomorphicvectorfield X in aut(M, p) such thatIm X isalsotangent (i.e. X is complextangent), then aut(M,p) is of infinite dimension [28]. We recall the followingdefinition.
Definition2.10.Areal-analytic hypersurface M ⊂ Cn+1is holomorphically
nondegener-ate at p ∈ M if thereisnogermat p of aholomorphicvectorfield X tangent to M . Denote by Θ the setof allrationalnumbersoftheform
q =
n
j=1
kjμj+ kn+1
forsomenonnegativeintegers k1, . . . , kn+1.
WedecomposetheformalTaylorexpansionof ψ, denotedby Ψ , into ΛM-homogeneous
polynomials ofweighteddegree ν, called Ψν, thatis,
Ψ =
ν∈Θ
Ψν.
Notice,using (2.4), that Ψν = 0,for ν < 1, and Ψ1= P .
Let h = (zj, w)∈ Aut(M,p). Weknow by[23] that h is oftheform(2.12),thatwe
rewriteas
zj= zj+ fj(z, w),
w= w + g(z, w), (2.13) whichtakes themultitypecoordinates(z, w) into themultitypecoordinates(z, w).
Putting f = (fj, . . . , fn),weconsider themappinggivenby
T = (f, g),
and, again,decompose eachpowerseries fj and g into ΛM-homogeneouspolynomialsof
weighted degree μ, called fj
μ and gμ, fj= μ∈Θ fjμ, g = μ∈Θ gμ.
Let v = ψ(z, ¯z, u) bethedefiningequation of M in thecoordinates(z, w),ofthe formgivenby(2.4),
ψz, ¯z, u= Pz, ¯z+ oΛM(1). (2.14)
Since h ∈ Aut(M,p), substituting(2.13)into v= ψ(z, z¯, u) weobtainthe transfor-mationformula
ψz + fz, u + iψ(z, ¯z, u), z + fz, u + iψ(z, ¯z, u), u +Re gz, u + iψ(z, ¯z, u)
= ψ(z, ¯z, u) +Im gz, u + iψ(z, ¯z, u). (2.15) Using (2.13),weonlyhavetoconsidertermsofweight μ ≥ 1 in(2.15).Weget
2Re n j=1 Pzj(z, ¯z)fjμ−1+μj z, u + iP (z, ¯z)=Im gμ z, u + iP (z, ¯z)+· · · , (2.16)
where dotsdenotetermsdepending on fj
ν−1+μj, gν, ψν,for ν < μ (there are nodotsif
μ = 1).
We are now inaposition to introducethe analogof theChern–Moser operator[10]
forpoints offinitemultitype.
Definition 2.11.Thegeneralized Chern–Moseroperator,denotedby L, isdefinedby
L(f, g) =Re igz, u + iP (z, ¯z)+ 2 n j=1 Pzj(z, ¯z)fj z, u + iP (z, ¯z) . (2.17)
Thefollowinglemmashowstherelationbetweenthekernelof L and theinfinitesimal CRautomorphismsofthemodelhypersurfacegivenby(2.5).(See[10]forthesameresult intheLevinondegeneratecase.)
Lemma 2.12. Let L be given by (2.17) and let (f, g) be given by (2.13). Then (f,g) lies in the kernel of L if and only if the vector field
Y =
n
j=1
fj(z, w)∂zj+ g(z, w)∂w
lies in aut(MP, p), where MP is given by(2.5).
Proof. Applying Y to v− P weobtain
Re Y (v − P )|MP =− 1 2Re igz, u + iP (z, ¯z)+ 2 n j=1 Pzj(z, ¯z)fj z, u + iP (z, ¯z) =−1 2L(f, g). 2 (2.18)
Wehavethefollowingproposition whichshowshowtoreducetheweightedjet deter-minationproblem fromAut(M, p) to aut(MP, p).
Proposition2.13. Let h = (z + f, w + g) ∈ Aut(M, p) be given by (2.13). Let
(f, g) =(f, g)μ, where (f, g)μ := f1μ−1+μ1, . . . , f n μ−1+μn, gμ .
Let μ0 be minimal such that (f,g)μ0 = 0.Then the (nontrivial vector) field
Y =
n
j=1
fjμ0−1+μj∂zj + gμ0∂w (2.19)
lies in aut(MP, p), where MP is given by(2.5).
Proof. Using(2.16)andthedefinitionof μ0, weobtainthat
L(f, g)μ0
= 0. Therefore,using Lemma 2.12,weobtainthat
Y =
n
j=1
fμj0−1+μj∂zj + gμ0∂w
belongstoaut(MP, p). Thisachieves theproof ofthetheorem. 2
Definition 2.14.Wesaythatthevectorfield Y = n j=1 Fj(z, w)∂zj + G(z, w)∂w
has homogeneous weight μ (≥ −1) if Fj is a weighted homogeneous polynomial of
weighted degree μ + μj, and G is ahomogeneouspolynomialofweighted degree μ + 1.
Theweightsintroduceanaturalgradingonaut(MP, p) in thefollowingsense.Writing
aut(MP, p) as
aut(MP, p) =
μ+1∈Θ
gμ,
wheregμconsistsofweightedhomogeneousvectorfieldsofweight μ, weobservethateach
weighted homogeneouscomponent Xμ ∈ gμ of X ∈ aut(MP, p) lies also inaut(MP, p).
Thereasonisthat v− P isweightedhomogeneous.
Gatheringallthepreviousresults,weobtainthefollowing proposition.
Proposition 2.15. Let M ⊂ Cn+1 be a smooth hypersurface of finite multitype
(m1, . . . , mn) given by (2.4). Let MP be the model hypersurface given by (2.5). Assume
that there exists μ0 such that
aut(MP, p) =
−1≤μ<μ0−1
gμ. (2.20)
Then any h = (z + f, w + g) ∈ Aut(M,p) given by (2.13) such that (f,g)μ = 0 for all
μ < μ0 is the identity map.
InthelightofProposition 2.15,weseethatinorderto studytheweightedjet deter-mination problem for Aut(M, p), it is enoughto study the weighted jet determination problem foraut(MP, p).
3. Rigidvectorfields
Inthissection,wedescribeanimportantclassofvectorfields X ∈ aut(MP, p), which
playacrucialroleinthestudyofaut(MP, p). Asbefore,let M⊂ Cn+1begivenby(2.4).
Definition 3.1.Let X be aholomorphicvectorfieldoftheform
X = n j=1 fj(z, w)∂ zj + g(z, w)∂w. (3.1)
Wesaythat X is rigidif f1, . . . , fn, g are allindependentofthevariable w.
Notethattherigidvectorfield W ,ofhomogeneousweight−1,given by
W = ∂w (3.2)
liesinaut(MP, p). Wewilldenoteby E the weightedhomogeneousvectorfieldofweight
0 definedby E = n j=1 μjzj∂zj + w∂w. (3.3)
E is the weighted Eulerfield. Notethat bythedefinition of μj, E is anonrigid vector
fieldlyinginaut(MP, p).
Lemma 3.2. Let X ∈ aut(MP, p) be a rigid holomorphic vector field. Suppose that X is
homogeneous of weight
ν >−μn=− min μj.
Then g = 0.
Proof. Since ν > −min μj, every fj = fj(z) in (3.1) is nonconstant. Hence, writing
(Re X)(Im w − P (z,z)) ¯ = 0 we see that every term involving fj is not pluriharmonic.
On the other hand, all terms involving g = g(z) are pluriharmonic, and hence can-not cancelthe former ones.Since g(z) is also nonconstant, we immediately obtainthe conclusion. 2
Wehavethefollowing theorem.
Theorem 3.3. Let MP be holomorphically nondegenerate, and let X ∈ aut(MP, p) be a
nonzero rigid vector field. Then all weighted homogeneous components of X have weight strictly less than one.
Proof. Write X = n j=1 fj(z)∂zj + g(z)∂w. Byassumption,wehave Re n j=1 fj(z)∂zj + g(z)∂w Im w − P (z, ¯z)= 0. (3.4) Identifyingweightedhomogeneous components,wemayassume,withoutlossof general-ity,that X is weighted homogeneous ofweight ν. Assume ν≥ 1 bycontradiction.Since
P (z, z) is ¯ weightedhomogeneousofweight1 andhasnopluriharmonicterms,itsterms haveweight < 1 in z. Thenextracting termsin(3.4)ofweight≥ 1 in z we obtain
n j=1 fj(z)∂ zj+ g(z)∂w Im w − P (z, ¯z)= 0. (3.5) Since MP is holomorphically nondegenerate, it follows that X = 0 contradicting the
assumption. 2
Wehavethefollowinglemma.
Lemma 3.4. Let X ∈ aut(MP, p) be a weighted homogeneous vector field, and let W ∈
aut(MP, p) be given by (3.2). There exists an integer l ≥ 1, and a rigid vector field
Y ∈ aut(MP, p) such that [[. . . [[X; W ]; W ]; . . .]; W ] = Y , where the string of brackets is
of length l.
Proof. Observethattheeffectoftakingthebracketof X with W is simplydifferentiation of thecoefficientwith respectto w. Alsonote that
Re[X; W ]v− P (z, ¯z)= [Re X, Re W ]v− P (z, ¯z). 2
Definition 3.5. We say that X ∈ aut(MP, p) is an l-integration of a rigid vector field
Y ∈ aut(MP, p) if thestringofbracketsdescribedintheabovelemmais oflength l.
Remark 3.6. By the above lemma, the general case will be reduced to the rigid case bytakingsufficiently manycommutatorswiththevectorfield W . Theproblemreduces then to
(i) describingrigidvectorfields;
(ii) analyzing towhatextentrigidfieldscan be“integrated”.
AsaconsequenceofTheorem 3.3,wecandividehomogeneousrigidvectorfieldsinto three types,andintroducethefollowing terminology.
Definition 3.7. Let X ∈ aut(MP, p) be arigid weighted homogeneousvector field. X is
called
(1) a shift if theweighted degreeof X is lessthanzero; (2) a rotation if theweighteddegreeof X is equaltozero;
(3) a generalized rotation if theweighteddegreeof X is biggerthanzeroand lessthan one.
NotethatintheLevinondegeneratecase(where P (z, z) ¯ = z,z isaquadraticform), generalizedrotationsdonotoccur.Infact,usingLemma 3.2andwriting(Re X) z, z= 0, weconclude X = 0.
The same fact holds in complex dimension two. Indeed, writing P (z, z) ¯ =
k≥k0ckz
kz¯m−k with c
k0 = 0, and X = zl∂z, the expansion of (Re X)P (z,z) has ¯ a
nonzerotermwith zk0z¯m−k0+l−1,andhencecannotbezero.
Ontheotherhand,incomplexdimension≥ 3,suchvectorfields dooccur: Example3.8.Take
P (z, ¯z) :=Re z1z¯2l, X := iz2l∂z1, l > 1,
wheretheweightsare μ1= μ2= l+11 .In[24],weshowedthattheLiealgebraof
infinites-imalautomorphisms hassixcomponents,
g = g−1⊕ g− 1
l+1 ⊕ g0⊕ g1−l+12 ⊕ g1−l+11 ⊕ g1. (3.6)
Inparticular,g1 isgeneratedbythe2-integrationof W
z1∂z1+ 1 lz2∂z2 w + 1 2w 2∂ w.
Proposition3.9. For ΛM and P as before, there exists a polynomial weighted homogeneous
change of coordinates such that the following hold:
(1) any rotation is linear;
(2) any non-transversal shift is of the form
X = j0 j=1 fj(z)∂zj + g(z)∂w (3.7) with fj0 = const = 0.
Proof. We performachangeofcoordinatesas inProposition 2.6.To show(1), observe thatinviewofLemma 3.2,anyrotationisoftheform
X = s,j λsjzs∂zj + α,j cα,jzα∂zj, (3.8)
where the weight of zα∂
zj iszero and αl = 0 whenever l ≥ j.It suffices to show that
all cα,j = 0.By contradiction,assume thatthere exist α and j with cα,j = 0. Among
thosechoose k to betheminimalpossible j.Since X is arotation,itisanautomorphism of MP,i.e.satisfies
(X + ¯X)P = 0. (3.9)
Considerthenontrivialmonomial czγkz¯γk
intheexpansionof P given byProposition 2.6. Expanding(3.9)weobtainintheleft-handsideanontrivialmonomialoftheform(2.7)
whichcontradicts (3.9), showing(1).
To show (2), assume that X is a non-transversal shift of the form (3.7) with no constant fj(z). Write X in theform
X =
α,j
cα,jzα∂zj + g∂w. (3.10)
Then choosinganonzeromonomial cα,jzα∂zj intheexpansionof X with minimal
pos-sible j and arguingas beforeweobtainacontradiction. 2 4. Integratingtransversal shifts
Inthissection,wefirstconsiderahomogeneous rigidvectorfield X∈ aut(MP),which
is ashift.Wecallit transversal, ifitisofweight−1, andhence
X = a∂w, a∈ R.
Wewillshow that X can beintegrated atmosttwotimes,provided MP is
holomorphi-cally nondegenerate.
Westartwiththefollowing definition.
Definition 4.1. We say that the weighted homogeneous holomorphic vector field n
j=1fj∂zj isa real reproducing fieldif
2Re
n
j=1
fj(z)Pzj(z, ¯z) = P (z, ¯z). (4.1)
Wesaythattheweighted homogeneousfieldnj=1fj∂
zj isa complex reproducing field
if 2 n j=1 fj(z)Pzj(z, ¯z) = P (z, ¯z). (4.2)
Thefollowing lemmaisstraightforward.
Lemma 4.2. The real reproducing fields are given by R + X, where
R =
n
j=1
μjzj∂zj (4.3)
and X is any rotation field.
Weneedthefollowinglemma.
Lemma 4.3. Let MP given by(2.5) be holomorphically nondegenerate, and let X, Y , U
be rigid holomorphic vector fields satisfying
(Re X)P = 0, (Re Y )P +Im(X + U)P2= 0, [U, X] = 0. (4.4)
Assume that X is of weight ≥ 0. Then X and Y commute.
Proof. SinceRe X commuteswithIm X,thefirstandthethirdequationsin(4.4)imply (Re X)Im(X + U)P2= 0. (4.5) HenceapplyingRe X tothesecondequationin(4.4),weobtain
(Re X)(Re Y )P = 0. (4.6) Ontheotherhand,thefirstequationin(4.4)implies
(Re Y )(Re X)P = 0. (4.7) From (4.6)and (4.7)weobtain
Re[X, Y ]P = 0, (4.8) andhence[X, Y ] is asymmetry.
Since X has weight bigger or equal to 0, it follows from (4.4) that Y has weight bigger or equalto 1. Hence[X, Y ] is asymmetry of weightbigger or equalto 1. Then
Theorem 3.3implies[X,Y ] = 0 asdesired. 2 Let X be afieldoftheform
X =
n
j=1
λjzj∂zj, λj ∈ C. (4.9)
It follows fromthediagonal formof X that everymonomial zα is aneigenvector of X
withtheeigenvalue
wX
zα:=λjαj. (4.10)
Wehavethefollowing result.
Lemma4.4. Let MP given by(2.5)be holomorphically nondegenerate, and let R be given
by (4.3), W = X + Z be a linear vector field in Jordan normal form with X of the form (4.9), and Z the nilpotent part, and λ ∈ R. Suppose that X, Y and U := Z + λR satisfy (4.4). Then wX+λR zα∈ R, 0 ≤ wX+λR zα≤ λ, (4.11)
for every nontrivial monomial czαz¯β in the expansion of P , where both inequalities are strict if λ = 0.
Proof. Notethatclearly U and X commute since X is ofweighteddegreezero,andsince
Z commutes with X. Wefirstshowthat
wX+λR
zα∈ R.
Assume by contradiction thatIm wX+λR(zα) = 0 forsome nontrivialmonomial czα¯zβ
intheexpansionof P . It iseasilyshown,usingthefactthat X is arotationthat
wX
zα+wX
zβ= 0, (4.12)
and hence,foreverynontrivialmonomial zαz¯β intheexpansionof P ,
wX+λR zα− wX+λR zβ= 2w X+λR zα− λ. (4.13) Wechooseareal linearfunction l : R→ R suchthat
lImwX+λR
zα− wX+λR
zβ> 0 (4.14)
for some monomial zαz¯β in the expansion of P . Let zα0z¯β0 be the minimal (in the
lexicographicorderingsense)nontrivialmonomialintheexpansionof P maximizing the left-handsideof(4.14).Then,usingtheJordan normalform,theexpansionof
Im(X + U)P2= 2P(Im X + U)P
containsthemonomial z2α0z¯2β0 with
lIm(wX+λR− wX+λR) z2α0z¯2β0> lIm(w X+λR− wX+λR) zα¯zβ (4.15) forallmonomials zαz¯βintheexpansionof P , thereasonbeing that(Re(X+λR))P = λP .
UsingLemma 4.3andthefactthat[R, Y ] = Y (Y isofweighteddegreeone),weconclude that Im wX+λR zγ∂zs zαz¯β=Im wX+λR ¯ zγ∂z¯s zαz¯β=Im wX+λR zαz¯β (4.16)
http://doc.rero.ch
for any monomial zαz¯β and therefore (Re Y )P contains only monomials ((Re(zγ∂ zs))zαz¯β) forwhich lIm(wX+λR− wX+λR) Rezγ∂zs zαz¯β = lIm(wX+λR− wX+λR) zαz¯β. (4.17)
Summarizing, we obtain thatthe second equation in(4.4), (4.15) and (4.17) together contradict(4.14) andtherefore
Im wX+λR
zα= 0. (4.18)
But(4.18)impliesthat(4.13)isantisymmetric withrespectto α and β. Weclaimthat
wX+λR
zα− wX+λR
zβ≤ λ (4.19)
whichwillimplybyantisymmetrythat
−λ ≤ wX+λR
zα− wX+λR
zβ≤ λ. (4.20)
First,assumebycontradictionthat
wX+λR
zα− wX+λR
zβ= λ + (4.21)
for some α and β in the expansion of P and > 0 being maximal possible. We shall chooseherethe minimalpossible α in the lexicographicorder.Usingagain Lemma 4.3
andthefactthat[R, Y ] = Y ,weobtainthat
wX+λR zγ∂zs zαz¯β= wX+λR zαz¯β+ λ, (4.22) wX+λR ¯ zγ∂¯zs zαz¯β= wX+λR zαz¯β (4.23) foranymonomial zαz¯βandtherefore(Re Y )P containsonlymonomialsRe(zγ∂
zs)(zαz¯β) forwhich (wX+λR− wX+λR) Rezγ∂zs zα¯zβ= (wX+λR− wX+λR) zαz¯β+ λ (4.24) or (wX+λR− wX+λR) Rezγ∂zs zαz¯β= (wX+λR− wX+λR) zαz¯β− λ. (4.25) Comparingtheweights inthesecond equation of(4.4), wesee thatthesecond termof theleft-handsideoftheequationcontainsanontrivialmonomialofweight2λ+2, > 0,
whilethefirsttermoftheequationhasweightatmost λ ++λ.Hencethecontradiction. Finally,if λ = 0,assumebycontradictionthat
wX+λR
zα− wX+λR
zβ= λ (4.26)
forsome α and β in theexpansionof P . Using(4.13),weobtainthat
wX+λR
zα= λ, wX+λR
zβ= 0. (4.27)
Let Pλ be the sum of all monomials in P for which (4.26) holds. Using the second
equation of(4.4)and(4.27),wesee that Pλ shouldsatisfy
(2λ +Im Z)Pλ2+ YPλ= 0. (4.28) But(4.28)cannot holdsince Pλcontainsnoharmonicterms.Indeed,takethenontrivial
monomial zαz¯β of Pλ with maximal |β| > 0 for which then (α,β) is minimal in the
lexicographicorder.Then(Pλ)2hasnontrivialmonomial z2αz¯2β whichcannotoccur in
Y Pλ andin(Im Z)(Pλ)2.Henceweobtainthecontradiction.Thisachievestheproofof
thelemma. 2
Proposition 4.5. Let MP given by (2.5) be holomorphically nondegenerate, R be given
by (4.3), W be a linear rotation, and Y be a rigid holomorphic vector field satisfying
(Re Y )P +Im(W + R)P2= 0. (4.29)
Then
Im(W + R)P = 0. (4.30) Weneedthefollowinglemma.
Lemma 4.6. Let W = X + Z be a linear vector field in Jordan normal form with X the diagonal and Z the nilpotent part. Assume that (Re W )P = 0. Then (Re X)P = (Re Z)P = 0.
Proof. Since X is diagonal,wehavethespectral decomposition
P =Pμ, Pμ∈ Pμ:=
zαz¯β: 2Re Xzαz¯β= μzαz¯β.
We consider thelexicographicorder on monomials zαz¯β. Weclaim thatP
μ = 0
un-less μ = 0. Indeed,assume by contradiction thatPμ = 0 for some μ = 0 and consider
the minimalnontrivialmonomial zαz¯β intheexpansionof Pμ with respect tothe
lexi-cographic order.Then Re Z(zαz¯β) has onlymonomials largerthan zαz¯β, thereforethe
coefficient of zαz¯β inRe W (P ) is equal to μ. Since Re W (P ) = 0,we must haveμ = 0
contradicting our assumption. Hence P = P0 as claimed, implying Re X(P ) = 0 and
henceRe W (P )= 0,whichimpliesRe Z(P )= 0. 2
Proofof Proposition 4.5. Afteralinear changeof(multitype) coordinates,we may as-sume that W = X + Z, with X the diagonaland Z the nilpotentpart.ByLemma 4.6,
X and Z are rotations.Nowdefine Pμ,0 < μ < 1, to bethesumofallmonomials zαzβ
of P with wX+R zα= μ. ByLemma 4.4, wehave P =Pμ.
Since P is real, the monomials zαz¯β and zβz¯α have conjugate coefficients. Since the
left-handsideof(4.13)isantisymmetric in α, β, theright-handsidesatisfies
wX+R zβz¯α= 1− wX+R zαz¯β. (4.31) Henceif zαz¯β enters P
μ, zβz¯α enters P1−μ.Thereforeusingrealityof P , wehave
Pμ= P1−μ. (4.32)
Incourseofproofweshallusetheconvention Pμ= 0 forany μ ≥ 1.
Then,identifyingtermsofweight μ in (4.29),weobtain ¯ Y Pμ= i(μ− 1) − Im Z ν PνPμ−ν, (4.33)
where μ − 1 < 0. Denoting T = i ¯Y we rewrite(4.33)as
T Pμ= (1− μ − i Im Z)
PνPμ−ν, 1− μ > 0. (4.34)
Withoutloss ofgenerality, P = 0.Set
l := min{μ: Pμ = 0} > 0. (4.35)
Then(4.34) implies
T Pl= 0. (4.36)
Conjugatingandusing(4.32)weobtain
¯
T P1−l= 0. (4.37)
Given thechoiceof l and using(4.32) weobtain
Pμ= 0, μ < l or μ > 1− l. (4.38)
In the sequel c1, c2, . . . , will always denote suitable positive integers. Consider the
(unique) integer s ≥ 1 satisfying
1− l ≤ sl < 1. (4.39)
Then applying s times T and using(4.34) and(4.38),weobtain
Ts−1Psl = (c1+ Q1)Pls, (4.40)
where Q1isa linearoperatorincreasingthelexicographicorderofmonomialsand
com-muting with T and T . ¯ Inthesequelweshallalways denoteby Q1, Q2, . . . , operatorsof
this kind.
Since Pl = 0, it follows from(4.39) and (4.38) that sl = 1 − l, since otherwise the
left-hand side vanishes and, choosing as before the minimal monomial of Pl we would
reachacontradiction.Hence1 isdivisibleby l.
Since T is holomorphic, it commutes with T . ¯ AlsoT and T commute ¯ with Im Z by
Lemma 4.3.Thenapplying T to¯ (4.40)andusing (4.37),weobtain
0 = (c1+ Q1)sPls−1T P¯ l, (4.41)
whichyields
¯
T Pl= 0 (4.42)
using thebynow frequentlyused argumentwith theminimal monomial. Weshall con-tinueusing thisargumentwithoutmentioningintherestof theproof.
Wenextclaimthat
Pμ = 0, l < μ < 2l. (4.43)
Indeed, otherwise take the minimum l < μ < 2l with Pμ = 0. Then using (4.38) we
obtain
0 = TsPsl−l+μ= (c2+ Q2)Pls−1Pμ (4.44)
and hence(4.43) holdsasdesired.
Now using(4.43)andapplying Ts−2 to Psl, weobtain
Ts−2Psl = (c4+ Q4)Pls−2P2l. (4.45)
Then applyingT and ¯ using(4.42)we conclude0= (c4+ Q4)Pls−2T P¯ 2l implying
¯
T P2l= 0. (4.46)
Next,similarlyto theclaim (4.43)asbefore,we prove
Pμ= 0, 2l < μ < 3l, (4.47)
wherewe repeatthepreviousarguments applying Ts−2 to Psl−2l+μ= 0.
Similarly,applying Ts fors = s− 2,s − 3, . . . , to P
sl+ε,0≤ ε < l, weconcludeby
induction
¯
T Pkl= 0 (4.48)
forall k and Pμ= 0 whenever μ is notdivisibleby l. Inparticular,weobtainT P¯ μ= 0 for
all μ and thereforeT P = 0. ¯ Finally, usingrealityof P , weconclude T P = 0 and hence ¯
Y P = 0 or, conjugating, Y P = 0. ThisimpliesthatIm(W + R)P = 0 asdesired. 2 Wemaynow statethemain resultofthissection.
Theorem 4.7. Let MP given by (2.5) be holomorphically nondegenerate, and consider
∂w∈ aut(MP, 0). Then there exists no vector field lying in aut(MP, 0) that is a
3-integra-tion of ∂w. Moreover, if we choose coordinates as in Proposition 3.9, every 1-integration
of ∂w is of the form
j
lj(z)∂zj + w∂w, (4.49)
where all lj are linear, and every 2-integration is of the form
j ϕj(z)w∂zj + 1 2w 2∂ w∈ aut(MP, 0), (4.50) where ϕj satisfy 2 j ϕj(z)Pzj = P (z, ¯z). (4.51)
Proof. If ∂w can beintegratedat leastonce,weobtainanautomorphism
j
ϕj(z)∂zj+
w + ϕ(z)∂w∈ aut(MP, 0), (4.52)
wheretheweightsof ϕj(z) arepositive.Applyingtwicetherealpartof(4.52)to P− v,
where w = u + iv,andputting w = u + iP weobtain 2Re
j
ϕj(z)Pzj − P (z, ¯z) − Im ϕ(z) = 0. (4.53)
Since thefirst two terms are all notpluriharmonic, we obtain ϕ(z) = 0.Therefore we have
2Re
j
ϕj(z)Pzj = P (z, ¯z). (4.54)
ByLemma 4.2,wecanwrite j ϕj(z)∂zj = R + W, (4.55) where R = n j=1 μjzj∂zj (4.56)
is theEuler vectorfieldand W is anyrotation(seeDefinition 3.7).
Assuming(4.52) canbe integrated,weobtainanewvectorfieldoftheform j ϕj(z)w + ψj(z) ∂zj+ 1 2w 2+ ψ(z) ∂w∈ aut(MP, 0). (4.57)
Applying twicetherealpartof(4.57)to P− v andputting w = u + iP weobtain
−2P (z, ¯z) Im j ϕj(z)Pzj + 2Re j Pzjψj(z)− Im ψ(z) = 0. (4.58)
Sincethefirsttwosummandscontainonlynon-pluriharmonicterms,weobtain ψ(z) = 0. Hence,weobtain −2P (z, ¯z) Im j ϕj(z)Pzj + 2Re j Pzjψj(z) = 0. (4.59)
ByProposition 3.9(1),wecanassumethat W is alinearrotation.Thenusing Propo-sition 4.5,weobtaininparticularthat
Im
j
ϕj(z)Pzj = 0. (4.60)
Then Theorem 3.3impliesthat ψj = 0,and hence,inviewof (4.54),(4.59)implies
2
j
ϕj(z)Pzj = P (z, ¯z). (4.61)
Assuming again(4.57) can furtherbe integrated,and using(4.61),we obtainafield of theform
Y = j 1 2w 2ϕ j(z) + χj(z) ∂zj+ 1 6w 3+ χ(z) ∂w∈ aut(MP, 0). (4.62)
Applyingtwicetherealpartof(4.62)to P− v,andusing(4.61),weobtain(with χ = 0 as above), Re1 2 u2− P2+ 2iuPP +Re j χj(z)Pzj − 1 6 3u2P− P3= 0. (4.63) Putting u = 0 in(4.63),weobtain −1 3P 3+ 2Re j χj(z)Pzj = 0. (4.64)
Multiplying(4.64)by P , weseethattheexpansionofthefirstexpressionhasnontrivial termsofweightedbidegree(2,2).Ontheotherhand,since Y is homogeneousofweight 2, theweightof χj(z) is2+ μj,andhencetheright-handsideof(4.64)cannothaveterms
ofweighted bidegree (2,2).We obtainacontradiction provingthatthe3-integration Y of ∂w cannotexist,provingthetheorem. 2
Equivalently,there exist weights Λ = (λ1, . . . , λn) (possiblydifferent frommultitype
weight), with respect to which P is diagonal, i.e. contains only monomials zαz¯β such
that|α|Λ=|β|Λ.
5. Integratingrotationsandgeneralizedrotations
In this section, we consider rotationsand generalized rotations.We show that they cannot beintegrated,providedthat MP isholomorphicallynondegenerate.
Wewrite P = l j=1 Pj, (5.1)
where Pj isasum ofmonomials of theformBαj, ˆαjzαjz¯αˆj of constantweighted length
|ˆαj|
ΛM =: cj,orderedsuchthat cj< ck for j < k. Westartwith thefollowing lemma.
Lemma5.1. Let X be a generalized rotation. Then there exists N > 0 such that XNP = 0.
Proof. Weassumethat X is ageneralizedrotationofweight ν > 0. Wethen have
X(P1) = 0, X(Pj) + X(Pk) = 0, (5.2)
where cj = ck+ ν, for everyk and Pj is given by(5.1). Since X and X commute, we
reachtheconclusion. 2
Weset thefollowing definition.
Definition 5.2.Let X be arigidholomorphicvectorfieldandlet p ∈ C[z,z]. ¯ Wedefine
dX(p) := sup
s + t: XsX¯t(p) = 0. (5.3) Remark5.3.UsingLemma 5.1,weseethat dX(P ) <∞ if X is ageneralizedrotationor
anilpotentlinearrotation.
Lemma5.4. Let MP given by(2.5)be holomorphically nondegenerate. Assume that X is
either a generalized rotation or a nilpotent linear rotation, and let Y be a rigid holomor-phic vector field satisfying
Re Y (P ) + Im XP2= 0. (5.4)
Then X = 0.
Proof. Assumebycontradictionthat X(P ) = 0.Let D be thesumofmonomialsofthe form Aα, ˆαzαz¯αˆ in P2forwhich|α|ΛM =|ˆα|ΛM.Weclaimthat
X(D) = 0.
Indeed,writing P as in(5.1),weobtainthat D can be writtenas
D = l j=1 PjPj. (5.5) Let ˆ dX(P ) := max j dX(Pj) , (5.6)
where dX(Pj) is given by (5.3). Let Pj
ˆ
d be the set of monomials of P
j of the form
Bαj, ˆαjzα j
¯
zαˆj forwhichthere exists s such that
XsX¯dˆZ(P )−sB
αj, ˆαjzα j
¯
zαˆj = 0.
Using Lemma 5.1and(5.5),we obtainthat
D = l j=1 Pj ˆ dP j ˆ d+ R, (5.7)
where XsX¯2 ˆdZ(P )−s(R)= 0,forevery s. But(5.7)impliesthat
dX(D) = 2 ˆdX(P ).
Since X(P ) = 0,wehavedˆX(P ) = 0,andhence dX(D) = 0.Since D is real,thisimplies
that X(D) = 0.Since X∈ aut(MP, 0),theassumption (5.4)implies
2XP2+ iRe Y (P ) = 0. (5.8) Since theweighted bidegree of D is (1,1),theweighted bidegree of X(D) is (1+ ν,1), where ν≥ 0 istheweightof X. Ontheother handeachterm inRe Y (P ) hasweighted bidegree (k, l) with eitherk < 1 or l < 1. Henceno termfrom2X(D) in(5.8) can get canceled byatermfromthesecondsummand.Since X(D) = 0,weobtainacontradiction with our assumption X(P ) = 0. Therefore X(P ) = 0, and hence X = 0 since MP is
holomorphicallynondegenerate. 2
Wemaynow statethemain resultofthissection.
Theorem 5.5. Let MP given by (2.5) be holomorphically nondegenerate, and let X ∈
aut(MP, 0) be either rotation or generalized rotation. There exists no vector field in
aut(MP, 0) that is a 1-integration of X.
Proof. Wewrite X as
X =
j
fj(z)∂zj. (5.9)
Recallthatanintegrationof X is anyvectorfield Y ∈ aut(MP, 0) satisfying[∂w, Y ] = X.
Then Y has tobe oftheform
Y = w j fj(z)∂zj+ j ϕj(z)∂zj+ ϕ(z)∂w∈ aut(MP, 0). Wethenhave 2Re Y (P − v) = Re 2 j Pzjfj(z) u + iP (z, ¯z)+ 2 j Pzjϕj(z) + iϕ(z) =Re 2 j Pzjfj(z)iP (z, ¯z) + j 2Pzjϕj(z) − Im ϕ(z) = 0,
wherewehaveusedRe X(P −v)= 0.Thefirsttwosumscontainonlynon-pluriharmonic terms,whilethelasttermispluriharmonic.It impliesthat ϕ(z) = 0 andhence
−P (z, ¯z) Im j Pzjfj(z) +Re j Pzjϕj(z) = 0. (5.10)
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Inthecase X is arotation,byProposition 3.9(1),afterapolynomialweighted homo-geneous change of coordinates, we may assume that X is linear. We may also assume thatthematrixof X is initsJordannormalform.ThenweconcludefromLemma 4.6, that thediagonal partof X is alsoa rotation.Applying Lemma 4.4, weconclude that the diagonalpart of X is zero.Therefore Lemma 5.4 together with(5.10) implies that
X = 0 contradicting theassumption.
On theother hand,if X is ageneralizedrotation,wecan directlyapply Lemma 5.4
together with(5.10) toconcludethat X = 0 contradicting theassumption.Theproofis complete. 2
We finishthis sectionby givinganother exampleof ahypersurfaceadmittinga gen-eralized rotation.
Example5.6. Let P be definedby
P (z, ¯z) =− Rez12z12z22
+ z12z2 2
, (5.11)
and MP by(1.6).Theweights are μ1= μ2 = 16. MP admitsthefollowing symmetryof
weight 1 6 Y = z1z2∂z1− z22∂z2. (5.12) Indeed,weobtain Y (v− P ) = −2z2z21z12z22+ 2z22z21z12z2 (5.13) hence Re Y (v − P ) = 0. (5.14) 6. Integratingnontransversalshifts
Inthissection,weshowthatnontransversalshiftscanbeintegratedatmostonetime. Westartwiththefollowing lemma.
Lemma 6.1. Let Z be a nontransversal shift on MP of the form
Z =
n
j=1
fj(z)∂zj + g(z)∂w. (6.1)
Then there exist modified multitype coordinates (with pluriharmonic terms allowed) such that there exists r, 1≤ r ≤ n, with
Z = i∂zr, (6.2)
and consequently,
Pyr(z, ¯z) = 0. (6.3)
Proof. We observe thatthere is j, 1≤ j ≤ n,such thatfj isnonzero, sinceotherwise
Z = 0.
We first assume that all μj are equal. It implies that all fj are constant. After a
possible holomorphiclinearchangeofcoordinates,wemayassumethat Z is givenby
Z = ∂z1+ g(z)∂w. (6.4)
Thefollowingchangeofmodifiedmultitype coordinatesleadstothedesiredconclusion
zj∗= zj,
w∗= w− z1g(z). (6.5)
Assumenowthatthe μj arenotallequal.Write
Z = jk j=1 fj(z)∂zj+ g(z)∂w, fjk = 0, (6.6) where μ1≥ · · · > μj1 =· · · = μjk.
Since Z = 0 isofnegativeweight, fjkisnonzeroand,inviewofProposition 3.9(2),can
beassumedtobeconstant.Afterperformingalinearchangeofthevariables zj1, . . . , zjk,
wemay assumethat k = 1 and fj1(z)= 1.
Thefollowingholomorphicchangeofcoordinates
z∗j = zj, 1≤ j ≤ j1− 2, zj∗1−1 = zj1−1− α Cα αj1+ 1 zjα1j1+1. . . z αjn jn , z∗j = zj, j1≤ j ≤ n, (6.7) where fj1−1(z)=C αz αj1 j1 . . . z αjn
jn ,leadstotheeliminationoftheterm f
j1−1(z)∂ zj1−1
in(6.6).
Similarlywecaneliminateany fj(z) with μ
j= μj1−1.Furthermore,usingrecursively
holomorphicchangesofcoordinatesasin(6.7),wecanarrange Z to becomeoftheform
Z = ∂zj1+ g(z)∂w. (6.8)
Finally, performing a change of coordinates similar to (6.5), we reach the desired conclusion.Thisachievestheproofofthelemma. 2
Assume now, according to Lemma 6.1, that MP admits, after a possible change of
modified multitypecoordinates,a nontransversalshift Z, givenby
Z = i∂zr. (6.9)
Wemay thenwrite P as
P (z, ¯z) = k j=0 xjrPj z, ¯z, Pk z, ¯z = 0, (6.10)
where z isthe(n− 1)-tupleof zj’swith zr omitted.Notethatif MP isholomorphically
nondegenerate, P must dependon zrandhence k≥ 1.
Theorem6.2. Assume that MP is holomorphically nondegenerate. Let Z be given by(6.9)
and P be given by (6.10). Then there is no 2-integration of Z.
Proof. Assuming Z can beintegrated,we obtainavectorfieldoftheform
wi∂zr+ n
j=1
ϕj(z)∂zj + ϕ(z)∂w∈ aut(MP, 0). (6.11)
Applyingtwicetherealpartof(6.11)to P− v,weobtain
2Re(u − iP )iPzr+ 2Re n
j=1
ϕj(z)Pzj − Im ϕ(z) = 0. (6.12)
Wemay rewrite(6.12),usingthehypothesisthat Z∈ aut(MP, 0),as
−P (z, ¯z) Im Z(P ) + Re n j=1 ϕj(z)Pzj + i 2ϕ(z) = 0. (6.13)
Assuming (6.11) can be integrated,and using(6.13),we obtainavector fieldof the form 1 2w 2i∂ zr + w n j=1 ϕj(z)∂zj + ϕ(z)∂w + n j=1 ψj(z)∂zj+ ψ(z)∂w∈ aut(MP, 0). (6.14)
Applying twicetherealpart(6.14)to P − v,we obtain
Reu2− P2+ 2iuPiPzr+Re(u + iP ) 2 n j=1 ϕj(z)Pzj+ iϕ(z) + 2Re n j=1 ψj(z)Pzj − Im ψ(z) = 0. (6.15)
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Putting u = 0 in(6.15),weobtain −P (z, ¯z) Im n j=1 ϕj(z)Pzj + i 2ϕ(z) +Re n j=1 ψj(z)Pzj + i 2ψ(z) = 0. (6.16)
Usingthehypothesis,wemay rewrite(6.13)and(6.16) as
−P Im Z(P ) + Re X(P − v) = 0, (6.17) −P Im X(P − v) + Re Y (P − v) = 0, (6.18) where X := n j=1 ϕj(z)∂zj+ ϕ(z)∂w, Y := n j=1 ψj(z)Pzj + ψ(z)∂w. (6.19)
Since Z = i∂zr,using (6.10)weobtain
k j=0 xjrPj z, ¯z k j=0 jxj−1r Pj z, ¯z − 2 Re X(P − v) = 0. (6.20) Similarly,rewriting(6.18)we have
k j=0 xjrPj z, ¯z Im X(P − v) + Re Y (P − v) = 0. (6.21) Weneedthefollowinglemma.
Lemma6.3. Let Pk be given by(6.10) and X be as above. Then
X(P − v) = Az, zzkrz¯rk+ k−1 l=1 Fl z, zzrk+lz¯rk−1−l+ F0 z, zzrkz¯rk−1 + F−1z, zzrk−1z¯rk+· · · , (6.22)
where the dots stand for lower degree terms with respect to the variables zr, zr, where
Fl z, z=−clPk2, l≥ 1, F0 z, z+ F−1z, z=−c0Pk2,
cl are positive coefficients and A(z, z) is purely imaginary.