MEROMORPHIC FUNCTIONS BIFURCATION SETS AND FIBRED LINKS

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MEROMORPHIC FUNCTIONS, BIFURCATION SETS AND FIBRED LINKS ARNAUD BODIN AND ANNE PICHON Abstract. We give a necessary condition for a meromorphic func- tion in several variables to give rise to a Milnor fibration of the local link (respectively of the link at infinity). In the case of two vari- ables we give some necessary and sufficient conditions for the local link (respectively the link at infinity) to be fibred. 1. Introduction A famous result of J. Milnor [11] states that the link f?1(0) ? S2n?1? (0 < ? 1) of a holomorphic germ f : (Cn, 0) ?? (C, 0) is a fibred link and moreover that a fibration is given by the so-called Milnor fibration f |f | : S 2n?1 ? \ f ?1(0) ?? S1. Throughout this paper Sn?1r denotes the sphere with radius r centered at the origin of Rn. The proof of this result has been extended in several directions in order to construct some natural fibrations in other situations of singu- larity theory. In this paper, we focus on two of them : (1) Let f : (Rn+k, 0) ?? (Rk, 0) be a real analytic germ with an isolated critical point at the origin. J. Milnor [11, Chapter 11] proved that for every sufficiently small sphere Sn+k?1? centered at the origin in Rn+k, the complement Sn+k?1? \ Lf of the link Lf = Sn+k?1? ?

condition called

without common

local fibrations

isolated

p1 such

milnor map

meromorphic functions

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profil-urra-2012

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MEROMORPHICFUNCTIONS,BIFURCATIONSETSANDFIBREDLINKSARNAUDBODINANDANNEPICHONAbstract.Wegiveanecessaryconditionforameromorphicfunc-tioninseveralvariablestogiverisetoaMilnorﬁbrationofthelocallink(respectivelyofthelinkatinﬁnity).Inthecaseoftwovari-ableswegivesomenecessaryandsuﬃcientconditionsforthelocallink(respectivelythelinkatinﬁnity)tobeﬁbred.1.IntroductionAfamousresultofJ.Milnor[11]statesthatthelinkf−1(0)∩Sε2n−1(0<ε1)ofaholomorphicgermf:(Cn,0)−→(C,0)isaﬁbredlinkandmoreoverthataﬁbrationisgivenbytheso-calledMilnorﬁbration|ff|:Sε2n−1\f−1(0)−→S1.ThroughoutthispaperSrn−1denotesthespherewithradiusrcenteredattheoriginofRn.Theproofofthisresulthasbeenextendedinseveraldirectionsinordertoconstructsomenaturalﬁbrationsinothersituationsofsingu-laritytheory.Inthispaper,wefocusontwoofthem:(1)Letf:(Rn+k,0)−→(Rk,0)bearealanalyticgermwithanisolatedcriticalpointattheorigin.J.Milnor[11,Chapter11]provedthatforeverysuﬃcientlysmallsphereSεn+k−1centeredattheorigininRn+k,thecomplementSεn+k−1\LfofthelinkLf=Sεn+k−1∩f−1(0)ﬁbresoverthecircle.AspointedoutbyMilnor,theﬁbrationisnotnecessarilygivenbytheMilnormapf.kfkThisresultcanbeextendedtoarealanalyticgermf:(X,p)−→(Rk,0)withisolatedcriticalvaluesatisfyingasuit-ablestratiﬁcationcondition,where(X,p)isagermofrealan-alyticspacewithisolatedsingularityatp([15],Theorem1.1).(2)Anotherdirectiondealswithlinksatinﬁnity.Letf:Cn−→Cbeapolynomialmap.Thelinkatinﬁnityassociatedwiththeﬁbref−1(0)isdeﬁnedbyLf,∞=f−1(0)∩S2Rn−1forasuﬃcientlylargeradiusR1.In[12],A.Ne´methiandA.ZahariaprovedDate:January29,2007.1

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