On the partial algebraicity of holomorphic mapping

arXiv:math/9906057v2[math.CV]6Apr2004ONTHEPARTIALALGEBRAICITYOFHOLOMORPHICMAPPINGSBETWEENTWOREALALGEBRAICSETSINCOMPLEXEUCLIDEANSPACESOFARBITRARYDIMENSIONJo¨elMerkerAbstract.TherigiditypropertiesofthelocalinvariantsofrealalgebraicCauchy-Riemannstructuresimposesuponholomorphicmappingssomeglobalrationalprop-erties(Poincar´e1907)ormoregenerallyalgebraicones(Webster1977).Ourprincipalgoalwillbetounifytheclassicalorrecentresultsinthesubject,buildingonastudyofthetranscendencedegree,todiscussalsotheusualassumptionofminimalityinthesenseofTumanov,inarbitrarydimension,withoutrankassumptionandforholomorphicmappingsbetweentwoarbitraryrealalgebraicsets.R´esum´e.Larigidit´edesinvariantslocauxdesstructuresdeCauchy-Riemannr´eellesalg´ebriquesimposeauxapplicationsholomorphesdespropri´et´esglobalesderatio-nalit´e(Poincar´e1907),ouplusg´en´eralementd’alg´ebricit´e(Webster1977).Notreobjectifprincipalserad’unifierlesr´esultatsclassiquesour´ecents,grˆace`aune´etudedudegr´edetranscendance,dediscuteraussil’hypoth`esehabituelledeminimalit´eausensdeTumanov,etceendimensionquelconque,sanshypoth`esederangetpourdesapplicationsholomorphesquelconquesentredeuxensemblesalg´ebriquesr´eelsar-bitraires.§1.IntroductionThealgebraicityortherationalityoflocalholomorphicmappingsbetweenrealalgebraicCRmanifoldscanbeconsideredtobeoneofthemostremarkablephe-nomenainCRgeometry.IntroducingtheconsiderationofSegrevarietiesinthehistoricalarticle[18],Webstergeneralizedtheclassicalrationalitypropertiesofself-mappingsbetweenthree-dimensionalspheresdiscoveredbyPoincar´eandlaterextendedbyTanakatoarbitrarydimension.Webster’stheoremstatesthatbi-holomorphismsbetweenLevinon-degeneraterealalgebraichypersurfacesinCnarealgebraic.Aroundtheeighties,someauthorsstudiedproperholomorphicmappingsbetweenspheresofdifferentdimensionsorbetweenpiecesofstronglypseudoconvexrealalgebraichypersurfaces,notablyPelles,Alexander,Fefferman,Pinchuk,Chern-Moser,Diederich-Fornaess,Faran,Cima-Suffridge,Forstneric,Sukhov,andothers(completereferencesareprovidedin[2,5,8,14,16,18,19]).Inthepastdecade,remov-ingtheequidimensionalityconditionintheclassicaltheoremofWebster,SukhovformappingsbetweenLevinon-degeneratequadrics[16],Huang[8]formappingsbe-tweenstronglypseudoconvexhypersurfaces,andSharipov-Sukhov[14]formappingsbetweengeneralLevinon-degeneraterealalgebraicCRmanifoldshaveexhibited1991MathematicsSubjectClassification.32V25,32V40,32V15,32V10.Keywordsandphrases.Holomorphicmappings,Realalgebraicsets,Transcendencedegree,Holomorphicfoliationsbycomplexdiscs,Algebraicdegeneracy,Zariski-genericpoint,MinimalityinthesenseofTumanov,Segrechains.TypesetbyAMS-TEX12JO¨ELMERKERvarioussufficientconditionsforthealgebraicityofagenerallocalholomorphicmapf:M→M′betweentworealalgebraicCRmanifoldsM⊂CnandM′⊂Cn′.Anecessaryandsufficientcondition,butwitharankconditiononfisprovidedin[2].Recently,usingpurelyalgebraicmethods,Coupet-Meylan-Sukhov([5];seealso[6])haveestimatedthetranscendencedegreeoffdirectly.Buildingontheirwork,weaimessentiallytostudythealgebraicityquestioninfullgenerality(cf.Problem2.4)andtounifythevariousapproachesof[2,5,8,12,14,18,19].Notably,weshallstatenecessaryandsufficientconditionsforthealgebraicityoffwithoutrankconditionandweshallstudythegeometryoftheminimalityassumptionthoroughly.§2.Presentationofthemainresult2.1.Algebraicityofholomorphicmappingsandtheirtranscendencede-gree.LetU⊂Cnbeasmallnonemptyopenpolydisc.Aholomorphicmap-pingf:U→Cn′,f∈H(U,Cn′),iscalledalgebraicifitsgraphiscontainedinanirreduciblen-dimensionalcomplexalgebraicsubsetofCn×Cn′.Usingclas-sicaleliminationtheory,onecanshowthat,equivalently,eachofitscomponentsg:=f1,...,fn′satisfiesanontrivialpolynomialequationgrar+···+a0=0,theaj∈C[z]beingpolynomials.WerecallthatasetΣ⊂UiscalledrealalgebraicifitisgivenasthezerosetinUofafinitenumberofrealalgebraicpolynomialsin(z1,...,zn,¯z1,...,¯zn).LetusdenotebyA=C[z]theringofcomplexpolynomialfunctionsoverCnandbyR=C(z)itsquotientfieldFr(A).ByR(f1,...,fn′)weunderstandthefieldgeneratedbyf1,...,fn′overR,whichisasubfieldofthefieldofmeromorphicfunctionsoverUandwhichidentifieswiththecollectionofrationalfunctions(2.2)R(f1,...,fn′)=P(f1,...,fn′)/Q(f1,...,fn′),P,Q∈R[x1,...,xn′],Q6=0.Following[5],thetranscendencedegree∇tr(f)ofthefieldR(f1,...,fn′)withrespecttothefieldRprovidesaninteger-valuedinvariantmeasuringthelackofalgebraicityoff.Inparticular,∇tr(f)iszeroifandonlyiffisalgebraic.Indeed,bydefinition∇tr(f)coincideswiththemaximalcardinalnumberκ′ofasubset{fj1,...,fjκ′}⊂{f1,...,fn′},1≤j1···jκ′≤n′whichisalgebraicallyindependentoverR.Inotherwords,∇tr(f)=κ′meansthatthereexistsasubset{fj1,...,fjκ′}⊂{f1,...,fn′}suchthattheredoesnotexistanontrivialrelationP(fj1,...,fjκ′)≡0inH(U),P∈R[x1,...,xκ′]\{0},butthatforeveryλ,κ′+1≤λ≤n′,every1≤j1···jλ≤n′,thereexistsanalgebraicrelationQ(fj1,...,fjλ)≡0,Q∈R[x1,...,xλ]\{0}.Ofcourse,∇tr(f1,...,fn′)≤n′.Anequivalentgeometriccharacterizationof∇tr(f)statesthat∇tr(f)=κ′ifandonlyifthedimensionoftheminimalforinclusioncomplexalgebraicsetΛf⊂U×Cn′containingthegraphΓf={(z,f(z))∈U×Cn′:z∈U}offisequalton+κ′(thiscomplexalgebra