CARTAN EQUIVALENCES FOR 5-DIMENSIONAL CR-MANIFOLDS IN C4 BELONGING TO GENERAL CLASS III 1 4 1 0 2 Masoud SABZEVARI andJoël MERKER n a J ABSTRACT. Wereducetovariousabsoluteparallelisms,namelytocer- tain e -structureson manifoldsof dimensions7, 6, 5, the biholomor- 7 { } 1 phic equivalence problem or the intrinsic CR equivalence problem for genericsubmanifoldsM5inC4ofCRdimension1andofcodimension ] 3thataremaximallyminimalandaregeometry-preservingdeformations V of one natural cubic model of Beloshapka, somewhere else called the C General Class III of 5-dimensional CR manifolds. Some inspiration 1 h. linksexistwiththetreatmentoftheGeneralClassIIpreviouslydonein t 2007byBeloshapka,Ezhov,Schmalz,andalsowiththeclassificationof a m nilpotentLiealgebrasduetoGoze,Khakimdjanov,Remm. [ 1 v 1. INTRODUCTION 7 9 ForsystematiccompletenessofourstudyofCR manifoldshaving: 2 4 dimension 6 5, . 1 0 in the case of a 3-dimensionalLevi nondegeneratehypersurface M3 C2, 4 ⊂ we intentionally re-constructed in [66] an explicit e -structure on a cer- 1 { } : tain natural 8-dimensional manifold N8 M3, because our current v −→ (wide)goalissimilarlytoperformcompletelyexplicitconstructionsof e - i X { } structuresforCR equivalencesin thesixGeneral Classes: r a I, II, III , III , IV , IV , 1 2 1 2 ofall the possiblyexistingembedded CR manifoldsup to dimension5 that werepresented in[55]. Thepresent memoirbeingspecificallydevotedto the: General Class III , 1 let us review at first the General Class I in order to appropriately recast the reader’s thoughtin therightaspirationto mathematicalunification. Date:2014-1-20. 2000 Mathematics Subject Classification. 32M05, 32V25, 32V35, 32V40, 53C10, 58A15,58A35,17B56,16W25. 1 2 MasoudSABZEVARI(Shahrekord)andJoëlMERKER(LM-Orsay) 1.1. Review of basic CR equivalences for Levi nondegenerate hy- persurfaces M3 C2. Let therefore M3 C2 be a (local) Levi- nondegenerate real⊂hypersurface of class at lea⊂st C6, graphed in coordi- nates: (z,w) = x+iy, u+iv as: (cid:0) (cid:1) v = ϕ(x,y,u). Anelementary normalization([52])insureswithoutlossofgeneralitythat: ϕ = x2 +y2 +O(3). Starting withthe(local)intrinsicgeneratorforT1,0M: ∂ ϕ ∂ L := z , ∂z − i+ϕ ∂u u havingconjugate: ∂ ϕ ∂ L = z , ∂z − i+ϕ ∂u u − whichinturn generates T0,1M = T1,0M, andintroducing: T := i L,L , onegets bytheLevinondegeneracya(cid:2)ssumpt(cid:3)ionthenatural frame: T ,L,L , forthecomplexifiedtangentbu(cid:8)ndleC (cid:9)TM. ⊗R We also introduce the dual coframe for C T∗M consisting of three ⊗R 1-formsdenoted: ρ , ζ , ζ , 0 0 0 namelysatisfyingby definition: (cid:8) (cid:9) ρ (T ) = 1 ρ (L) = 0 ρ (L) = 0, 0 0 0 ζ (T ) = 0 ζ (L) = 1 ζ L = 0, 0 0 0 ζ (T ) = 0 ζ (L) = 0 ζ L = 1. 0 0 0(cid:0) (cid:1) Abbreviatingnext: (cid:0) (cid:1) ϕ z A := , − i+ϕ u sothat: ∂ ∂ ∂ ∂ T = i +A , +A , ∂z ∂u ∂z ∂u (cid:20) (cid:21) theLevi-factorfunction: ℓ := i A +AA A AA , z u z u − − (cid:0) (cid:1) 1.Introduction 3 occurringin: ∂ T = ℓ ∂u isthereforenowhere vanishingby Levinondegeneracy. Onealso computesnext: ∂ ∂ ∂ L, T = +A , ℓ ∂z ∂u ∂u (cid:20) (cid:21) (cid:2) (cid:3) ∂ = ℓ +Aℓ ℓA z u u − ∂u (cid:16)ℓ +Aℓ ℓA (cid:17) = z u − u T . ℓ Thentheappearing function: ℓ +Aℓ ℓA z u u P := − , ℓ happens to be the single one which enters the so-called initialDarboux structure: dρ = P ρ ζ +P ρ ζ +iζ ζ , 0 0 ∧ 0 0 ∧ 0 0 ∧ 0 dζ = 0, 0 dζ = 0. 0 As explained in [53, 66] and as is quite also very well known, the initial ambiguity matrix group for (local) biholomorphic or CR equivalences of suchhypersurfaces is: aa 0 0 b a 0 GL (C): a C, b C , 3 ( b 0 a ! ∈ ∈ ∈ ) just because through any extrinsic local biholomorphism (or through any intrinsiclocal CR-equivalence): h: M M′, −→ onehas ([53])fora certain coefficient-functiona: h′ L′ = aL, ∗ h′(cid:0)L′(cid:1) = aL, ∗ (cid:0) (cid:1) 4 MasoudSABZEVARI(Shahrekord)andJoëlMERKER(LM-Orsay) whence: h′ T ′ = h′ i L′, L′ ∗ ∗ (cid:0) (cid:1) = i h(cid:0)′(cid:2)L′ , h′(cid:3)(cid:1)L′ ∗ ∗ = i(cid:2)aL(cid:0) , a(cid:1)L (cid:0) (cid:1)(cid:3) = a(cid:2)a i L, L (cid:3)+iaL a L iaL(a) L, · − · (cid:2) =T (cid:3) =:b(cid:0) (cid:1) =:b sothatsetting: | {z } | {z } | {z } b := iaL(a), − forgettinghowthisfurthercoefficient-function isrelated toa, oneobtains: h′ T ′ = aaT +bL +bL. ∗ Cartan’s gististo deal(cid:0)wit(cid:1)htheso-calledliftedcoframe: ρ aa 0 0 ρ 0 ζ := b a 0 ζ , 0 ζ b 0 a ζ 0 inthespaceof x,y,u,a,a,b,b . In [66], aftertwoabsorbtions-normalizationsand afteroneprolongation, (cid:0) (cid:1) the desired equivalence problem transforms to that of some — explicitly computed — eight-dimensionalcoframe: ρ,ζ,ζ,α,β,α,β,δ onacertain manifoldN8 (cid:8) M3 having e -s(cid:9)tructureequations: −→ { } dρ = α ρ+α ρ+iζ ζ, ∧ ∧ ∧ dζ = β ρ+α ζ, ∧ ∧ dζ = β ρ+α ζ, ∧ ∧ dα = δ ρ+2iζ β +iζ β, ∧ ∧ ∧ dβ = δ ζ +β α+Iζ ρ, ∧ ∧ ∧ dα = δ ρ 2iζ β iζ β, ∧ − ∧ − ∧ dβ = δ ζ +β α+Iζ ρ, ∧ ∧ ∧ dδ = δ α+δ α+iβ β +Tρ ζ +Tρ ζ, ∧ ∧ ∧ ∧ ∧ withthesingleprimary complexinvariant: 1 1 I := 2L L L P +3L L L P 7P L L P + 6 aa3 − − (cid:16) +4P L(cid:0) L(cid:0) P(cid:0) (cid:1)(cid:1)(cid:1)L P (cid:0)L (cid:0)P (cid:0)+2(cid:1)(cid:1)P(cid:1)P L P (cid:0), (cid:0) (cid:1)(cid:1) − (cid:17) (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 1.Introduction 5 andwithonesecondary invariant: 1 b T = L(I) P I i I. a − − aa (cid:18) (cid:19) 1.2. Explicitness obstacle. At the level of the function P, the completely explicit formulas for I, for T and for the 1-forms constituting the e - { } structure remain writable on an article, but not so anymore when one ex- presseseverythingbackinterms ofthegraphingfunctionϕ. Indeed, thereal and imaginaryparts∆ and ∆ in: 1 2 4 I = ∆ +i∆ aa3 1 4 havenumeratorscontainingrespect(cid:0)ively([58]):(cid:1) 1553198 and 1634457 monomialsin thedifferentialring in 6+3 1 = 83variables: 3 − Z ϕx,ϕy,ϕx2,ϕy2,ϕu2,ϕxy,ϕxu(cid:0),ϕy(cid:1)u,......,ϕx6,ϕy6,ϕu6,... . Hencecontrarytothegeneralcasewhereϕ = ϕ(x,y,u)doesdependupon (cid:2) (cid:3) the‘CR-transversal’variableu,intheso-calledrigidcase(oftenusefulasa caseofstudy-exploration)where ϕ = ϕ(x,y)isindependentofuso that: ϕ zzz P = , ϕ zz onerealizes thatIisrather easilywritable: 1 1 ϕ ϕ ϕ ϕ ϕ ϕ ϕ I = z2z4 6 z2z3 zz2 zz4 z2z 4 zz3 z2z2+ 6 aa3 ϕ − (ϕ )2 − (ϕ )2 − (ϕ )2 (cid:12)rigid (cid:18) zz zz zz zz case (cid:12) (cid:12) ϕ ϕ ϕ (ϕ )2ϕ (ϕ )3ϕ (cid:12) +10 zz3 z2z zz2 +15 zz2 z2z2 15 zz2 z2z , (ϕ )3 (ϕ )3 − (ϕ )4 zz zz zz (cid:19) and this therefore shows that there is a tremendous explosion of computa- tionalcomplexitywhen onepassesfromtherigidcaseto thegeneralcase. Consequently,oneexpectsanevenmuchdeepercomputationalcomplex- ity when one addresses the question of passing to CR manifolds of higher dimensions. 1.3. Embedded CR submanifolds ofCR dimension1 andnilpotent Lie algebras. Consider now a general sufficiently smooth generic submani- fold: M2+d C1+d ⊂ having CR dimension 1 and real codimension d > 1. According to the backgroundarticle[55], thecore bundleis: T1,0M := T1,0C1+n C TM . ∩ ⊗R (cid:0) (cid:1) 6 MasoudSABZEVARI(Shahrekord)andJoëlMERKER(LM-Orsay) To simplify (in fact, just a bit) the mathematical discussions, we shall as- sumethroughoutthatM isreal analytic. The question is: to understand the possible initial geometries of such CR manifolds, at least at a Zariski-generic point. This question is not yet completelysolved,becauseitopensupinfinitelymanybranchesofclassifi- cation. Yet, one can present well known general considerations which were al- readytransparentlyexplainedinSophusLie’soriginalwritings(cf. e.g.[19, 20, 21]), thoughnottargetlyin aCR context. IntroducethesubdistributionsofC TM: ⊗R D1M := T1,0M +T0,1M, C D2M := Span D1M + T1,0M, D1M + T0,1M, D1M , C Cω(M) C C C (cid:16) (cid:17) (cid:2) (cid:3) (cid:2) (cid:3) D3M := Span D2M + T1,0M, D2M + T0,1M, D2M , C Cω(M) C C C (cid:16) (cid:17) andgenerally: (cid:2) (cid:3) (cid:2) (cid:3) Dk+1M := Span DkM + T1,0M, DkM + T0,1M, DkM . C Cω(M) C C C (cid:16) (cid:17) BypassingtosomeappropriateZari(cid:2)ski-opensubse(cid:3)tof(cid:2)M,onemayass(cid:3)ume as isknownthat alltheseDkM becometruecomplexvector subbundlesof C TM havingincreasingranks: ⊗R 2 = r (M) < r (M) < < r (M) = r (M) = r (M) = , 1 2 ··· kM kM+1 kM+2 ··· until a first and final stabilization occurs. So we will admit that the M we consider enjoy constancies of such ranks, and of several other invariants (finiteinnumber)which mighthappen topop uplateron. As isknowntoo,intheveryspecial circumstancewhere: 2 = r (M) = r (M) = r (M) = , 1 2 3 ··· namelywhen: T1,0M, T0,1M T1,0M +T0,1M, ⊂ thereal analyticCR(cid:2)-generic subma(cid:3)nifoldM2+d C1+d is, locallyinsome ⊂ smallneighborhoodofeachofitspoints,biholomorphictoC Rd,adegen- × erate case rapidly set away. Sometimes, one also says that M is Levi-flat, orequivalently(justwhen theCR dimensionequals1), that M isholomor- phicallydegenerate([42]). In fact, wheneverthemaximalpossiblerank: r (M) = r (M) = r (M) = < 2+d = dim M, kM kM+1 kM+2 ··· R 1.Introduction 7 is still smaller than the dimension of M (not necessarily equal to 2), one realizes that M is similarly, locally in a neighborhood of a Zariski-generic point, biholomorphic to a CR-generic submanifold of C1+d which is con- tained in 2 + d r (M) straight real hyperplanes in transverse intersec- − kM tion,acasewhichisalsodegeneratehenceisalsodisregarded,justbecause it essentially comes down to the case of CR-generic submanifolds having smallerdimensionthat2+d. So the question is: to understand all the possible geometries of such CR manifoldsM2+d C1+dofCRdimension1thathavetheso-calledconstant ⊂ nonholonomicproperty that: DkMM = C TM. C ⊗R In the Several Complex Variables literature, such CR manifolds happen to be those called minimal in the sense of Tumanov, or equivalently offinite typeinthesenseofBloom-Graham,andtheyhappen tonecessarilybealso simultaneously holomorphically nondegenerate too (an implication which istrueonlyinCR dimension1, as iseasilychecked). Beloshapka and his students, e.g. Shananina, Mamai, Kossovskiy and others, have put some emphasis on the study of such a class of CR mani- folds, notably in the search for nice models which would potentiallyreveal newCartan geometries. The truth is that this research field, like the one of hyperbolic groups in the sense of Gromov, is per se opened to infinitely many untamable branches of complexity, for one soon realizes after a moment of reflection thattheLiealgebrasofinfinitesimalCRautomorphisms(assumingforsim- plicitythateverythingisrealanalytic)aredeeplyrelatedtotheclassification of nilpotent Lie algebras, an area which is known to be very rich and very infinite, as Lie himself understood more than one century ago (see Chap- ter 28 in the English translation [47] of VolumeI of theTheoriederTrans- formationsgruppen). Section 6 here is devoted to review the easiest part (only up to dimension 5) of the deep nilpotent Lie algebra classification theorems ofGoze, Khakimdjanov,Remm up to dimension8 ([29, 30, 31]), whichalready showsupan explodingramificationofvery manybranches. In dimension4, thereisasingleirreduciblenilpotentLiealgebra: [x ,x ] = x , 1 2 3 n1: 4 [x ,x ] = x . (cid:26) 1 3 4 Correspondingly, as Beloshapka discovered in 1997 ([4]), a real analytic 4-dimensional local CR-generic submanifold M4 C3 of codimension 2 ⊂ 8 MasoudSABZEVARI(Shahrekord)andJoëlMERKER(LM-Orsay) whosecomplextangentbundlesatisfies: C TM = T1,0M +T0,1M +[T1,0M, T0,1M]+ T1,0M,[T1,0M, T0,1M] + ⊗R + T0,1M,[T1,0M, T0,1M] (cid:2) (cid:3) mayalways berepresented, insuitableholomorphiccoordinates: (cid:2) (cid:3) z,w ,w = z,u +iv ,u +iv 1 2 1 1 2 2 bytwocomplexdefiningequationsofthespecificform: (cid:0) (cid:1) (cid:0) (cid:1) v = zz +O (x,y,u ,u ), 1 4 1 2 v = zz z +z +O (x,y,u ,u ). 2 4 1 2 Since then, such CR manifolds have been intensively studied, by (cid:0) (cid:1) Beloshapka-Ezhov-Schmalz who constructed a canonical Cartan con- nection ([9]) and who generalized Pinchuk-Vitushkin’s germ extension phenomenon ([8]), by Gammel-Kossovskiy ([24]), and by Beloshapka- Kossovskiywho providedafinal completeclassification([10]). In [55], onerefers to the: General Class II: M4 C3 with L, L, L,L , L, L,L ⊂ constintutingafra(cid:2)mefor(cid:3)C(cid:2) T(cid:2)M. (cid:3)(cid:3)o ⊗R Thenextnatural General Class ([55])isthe: General Class III : 1 M5 C4 with L, L, L,L , L, L,L , L, L,L ⊂ constintutingafra(cid:2)mefor(cid:3)C(cid:2) T(cid:2)M, (cid:3)(cid:3) (cid:2) (cid:2) (cid:3)(cid:3)o ⊗R and it is intrinsically related to the irreducible 5-dimensional nilpotent Lie algebra(labeled hereinthenotationofGoze-Remm): [x ,x ] = x , 1 2 3 n4: [x ,x ] = x , 5 1 3 4 [x ,x ] = x . 2 3 5 Three years ago, we started to study CR equivalences of such CR man- ifolds belonging to the General Class III , trying in the first months to di- 1 rectlyconstructaCartan connectionas didBeloshapka-Ezhov-Schmalzfor theGeneralClass II. ButinspiredbyChern’s seminal1939paperonequiv- alences of third order ordinary differential equations under contact trans- formations, we realized that it would be better to perform at first a pure exploration of the problem by employing the powerful tools of Cartan’s 1.Introduction 9 method of equivalence, in order to avoid as much as possible those errors of understanding that are caused by a too quick belief that certain features wouldsomewhateasilygeneralize. Becausewehavenotbeensincethenawareofanyotherpaperorpreprint or author having attacked the same problem, we decided to wait until the studyreachedapointofmaturityinwhicheverythingcouldbepresentedin fullcomputationaldetails,whatevercomplexitythetheory has. Of course in such a General Class III , it is known that the cubic model 1 M5 C4 incoordinates: c ⊂ z,w ,w ,w = z, u +iv , u +iv , u +iv 1 2 3 1 1 2 2 3 3 wasalso dis(cid:0)coveredby Be(cid:1)losh(cid:0)apka: (cid:1) v = zz, 1 v = zz z +z , 2 v = zz iz +iz . 3 (cid:0) (cid:1) − ButtheCartaninvariantsofthegeo(cid:0)metry-preser(cid:1)vingdeformationsofsucha modelhaveapparentlyneverbeenstudied,andsuchastudyisthemaingoal of the present memoir. Granted that the general class IV is already well 1 studied since Chern-Moser ([15]), a forthcoming paper by Samuel Pocchi- olawillsoontreattheGeneral ClassIII (as presentedin[55]), thusclosing 2 up the study of CR equivalences of CR manifolds up to dimension 5 (an overallsystematicreviewisplannedto appearat theend). With J being the standard complex structure of TC4, one sets as usual ([55]): TcM := TM J(TM), ∩ orequivalently: TcM := ReT1,0M. Ourfirst elementary resultappears in Section 5, cf. also [52]. Proposition 1.1. Every real analytic 5-dimensional local CR-generic sub- manifold M5 C4 of codimension 3 which is maximally minimal, namely ⊂ whichsatisfies: D1M = TcM hasrank2, D2M = TcM +[TcM,TcM] hasrank3, D3M = TcM +[TcM,TcM]+ TcM,[TcM,TcM] hasmaximalpossiblerank5, (cid:2) (cid:3) 10 MasoudSABZEVARI(Shahrekord)andJoëlMERKER(LM-Orsay) mayberepresented,insuitableholomorphiccoordinates(z,w ,w ,w ),by 1 2 3 threecomplexdefiningequationsofthespecificform: w w = 2izz +Π z,z,w ,w ,w , 1 1 1 1 2 3 − w w = 2izz(z +z)+Π z,z,w ,w ,w , 2 2 (cid:0) 2 1(cid:1) 2 3 − w w = 2zz(z z)+Π z,z,w ,w ,w , 3 − 3 − 3 (cid:0) 1 2 3 (cid:1) wherethethreeremaindersΠ , Π , Π area(cid:0)llan O( z 4)+(cid:1)zzO( w ). 1 2 3 | | | | Conversely,foranychoiceofthreesuchanalyticfunctionsenjoyingthese conditions,thezero-locusofthethreeequationsaboverepresentsarealan- alytic 5-dimensional local CR-generic submanifold M5 C4 of codimen- ⊂ sion3which ismaximallyminimal. (cid:3) Next,ageneral (1,0)holomorphicvectorfield: ∂ ∂ ∂ ∂ X = Z(z,w) +W1(z,w) +W2(z,w) +W3(z,w) ∂z ∂w ∂w ∂w 2 2 3 isaninfinitesimalCRautomorphismofBeloshapka’scubicmodelM5 ifby c definitionX+X is tangent to M5. By analyzing ingreat detailsthe system c oflinearpartialdifferentialequationssatisfiedbytheunknownfunctionsZ, W1,W2,W3, weobtainthesecond already known: Proposition 1.2. The Lie algebra aut (M) = 2Rehol(M) of the in- CR finitesimal CR automorphisms of the 5-dimensional 3-codimensional CR- genericmodel cubicM5 C4 represented bythethreereal graphedequa- c ⊂ tions: w w = 2izz, 1 1 − w w = 2izz(z +z), 2 − 2 w w = 2zz(z z), 3 3 − − is 7-dimensional and itis generated by the R-linearly independent real partsof thefollowingseven(1,0) holomorphicvectorfields: T := ∂ , w1 S := ∂ , 1 w2 S := ∂ , 2 w3 L := ∂ +(2iz)∂ +(2iz2 +4w )∂ +2z2∂ , 1 z w1 1 w2 w3 L := i∂ +(2z)∂ +(2z2)∂ (2iz2 4w )∂ , 2 z w1 w2 − − 1 w3 D := z∂ +2w ∂ +3w ∂ +3w ∂ , z 1 w1 2 w2 3 w3 R := iz∂ w ∂ +w ∂ , z − 3 w2 2 w3 havingLiebracketcommutatortable: