MILNOR FIBRATIONS OF MEROMORPHIC FUNCTIONS

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Niveau: Supérieur, Doctorat, Bac+8
MILNOR FIBRATIONS OF MEROMORPHIC FUNCTIONS ARNAUD BODIN, ANNE PICHON, JOSE SEADE Abstract. In analogy with the holomorphic case, we compare the topology of Milnor fibrations associated to a meromorphic germ f/g : the local Milnor fibrations given on Milnor tubes over punc- tured discs around the critical values of f/g, and the Milnor fibra- tion on a sphere. 1. Introduction The classical fibration theorem of Milnor in [6] says that every holo- morphic map (germ) f : (Cn, 0) ? (C, 0) with n > 2 and a critical point at 0 ? Cn has two naturally associated fibre bundles, and both of these are equivalent. The first is: (1) ? = f |f | : S? \K ?? S1 where S? is a sufficiently small sphere around 0 ? Cn and K = f?1(0)? S? is the link of f at 0. The second fibration is: (2) f : B? ? f?1(∂D?) ?? ∂D? ?= S1 where B? is the closed ball in Cn with boundary S? and D? is a disc around 0 ? C which is sufficiently small with respect to ?. The set N(?, ?) = B??f?1(∂D?) is usually called a local Milnor tube for f at 0, and it is diffeomorphic to S? minus an open regular neigh- bourhood T of K.

  • truncated global

  • milnor's proof concerns

  • meromorphic function

  • see also

  • local milnor

  • global milnor

  • milnor

  • lf ?


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MILNORFIBRATIONSOFMEROMORPHICFUNCTIONSARNAUDBODIN,ANNEPICHON,JOSE´SEADEAbstract.Inanalogywiththeholomorphiccase,wecomparethetopologyofMilnorfibrationsassociatedtoameromorphicgermf/g:thelocalMilnorfibrationsgivenonMilnortubesoverpunc-tureddiscsaroundthecriticalvaluesoff/g,andtheMilnorfibra-tiononasphere.1.IntroductionTheclassicalfibrationtheoremofMilnorin[6]saysthateveryholo-morphicmap(germ)f:(Cn,0)(C,0)withn>2andacriticalpointat0Cnhastwonaturallyassociatedfibrebundles,andbothoftheseareequivalent.Thefirstis:f(1)φ=:Sε\K−→S1|f|whereSεisasufficientlysmallspherearound0CnandK=f1(0)Sεisthelinkoffat0.Thesecondfibrationis:(2)f:Bεf1(Dδ)−→Dδ=S1whereBεistheclosedballinCnwithboundarySεandDδisadiscaround0Cwhichissufficientlysmallwithrespecttoε.ThesetN(ε,δ)=Bεf1(Dδ)isusuallycalledalocalMilnortubeforfat0,anditisdiffeomorphictoSεminusanopenregularneigh-bourhoodTofK.(Thus,togettheequivalenceofthetwofibrationsonehasto“extend”thelatterfibrationtoT\K.)Infact,inordertohavethesecondfibrationoneneedstoknowthateverymap-germfasabovehastheso-called“Thomproperty”,whichwasnotknownwhenMilnorwrotehisbook.Whatheprovesisthatthefibersin(1)arediffeomorphictotheintersectionf1(t)Bεfortcloseenoughto0.Thestatementthat(2)isafibrebundlewasprovedlaterin[5]byLeˆDate:February16,2009.2000MathematicsSubjectClassification.14J17,32S25,57M25.Keywordsandphrases.Meromorphicfunctions,Milnorfibration,Semitamemaps,(i)-tame.1
2ARNAUDBODIN,ANNEPICHON,JOSE´SEADEDu˜ngTra´nginthemoregeneralsettingofholomorphicmapsdefinedonarbitrarycomplexanalyticspaces,andwecallittheMilnor-Leˆfi-brationoff.Onceweknowthat(2)isafibrebundle,theargumentsof[6,Chapter5]showthisisequivalenttotheMilnorfibration(1).Theliteratureaboutthesefibrationsisvast,andsoaretheirgen-eralizationstovarioussettings,includingrealanalyticmap-germsandmeromorphicmaps,andthatisthestartingpointofthisarticle.LetUbeanopenneighbourhoodof0inCnandletf,g:U−→Cbetwoholomorphicfunctionswithoutcommonfactorssuchthatf(0)=g(0)=0.LetusconsiderthemeromorphicfunctionF=f/g:UCP1definedby(f/g)(x)=[f(x)/g(x)].Asin[3],twosuchgermsat0,F=f/gandF0=f0/g0areconsideredasequal(orequivalent)ifandonlyiff=hf0andg=hg0forsomeholomorphicgermh:CnCsuchthath(0)6=0.Noticethatf/gisnotdefinedonthewholeU;itsindeterminationlocusis I=zU|f(x)=0andg(x)=0.Inparticular,thefibersofF=f/gdonotcontainanypointofI:foreachcC,thefiberF1(c)istheset F1(c)=xU|f(x)cg(x)=0\I.Inaseriesofarticles,S.M.Gusein-Zade,I.LuengoandA.Melle-Herna´ndez,andlaterD.SiersmaandM.Tibaˇr,studiedlocalMilnorfibrationsofthetype(2)associatedtoeverycriticalvalueofthemero-morphicmapF=f/g.Seeforinstance[3,4],orTibar’sbook[12]andthereferencesinit.Ofcoursethe“Milnortubes”BεF1(Dδ)inthiscasearenotactualtubesingeneral,sincetheymaycontain0Uintheirclosure.Theseareinfact“pinchedtubes”.Itisthusnaturaltoaskwhetheronehasformeromorphicmap-germsfibrationsofMilnortype(1),andifso,howthesearerelatedtothoseoftheMilnor-Leˆtype(2)studied(forinstance)in[3,4,12].Thefirstofthesequestionswasaddressedin[10,1,11]fromtwodifferentviewpoints,whiletheanswertothesecondquestionisthebulkofthisarticle.Infact,itisprovedin[1]thatifthemeromorphicgermF=f/gissemitame(seethedefinitioninSection2),theng/fF1(3)|F|=|f/g|:Sε\(LfLg)−→Sisafiberbundle,whereLf={f=0}∩SεandLg={g=0}∩Sεaretheorientedlinksoffandg.NoticethatawayfromthelinkLfLg