MEROMORPHIC FUNCTIONS BIFURCATION SETS AND FIBRED LINKS

Description

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

Sujets

• 4799 Hirasawa
• Rudolph
• Milnor map
• Meromorphic function

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 13
• Langue: English

Extrait

MEROMORPHICFUNCTIONS,BIFURCATIONSETS ANDFIBREDLINKS ARNAUDBODINANDANNEPICHON Abstract. Wegiveanecessaryconditionforameromorphicfunc- tioninseveralvariablestogiverisetoaMilnorfibrationofthelocal link(respectivelyofthelinkatinfinity).Inthecaseoftwovari- ableswegivesomenecessaryandsufficientconditionsforthelocal link(respectivelythelinkatinfinity)tobefibred. 1. Introduction AfamousresultofJ.Milnor[11]statesthatthelinkf − 1 (0) ∩ S ε 2n − 1 (0<ε  1)ofaholomorphicgermf:( C n ,0) −→ ( C ,0)isafibredlink andmoreoverthatafibrationisgivenbytheso-calledMilnorfibration | f f | : S ε 2n − 1 \ f − 1 (0) −→ S 1 .Throughoutthispaper S r n − 1 denotesthe spherewithradiusrcenteredattheoriginof R n . Theproofofthisresulthasbeenextendedinseveraldirectionsin ordertoconstructsomenaturalfibrationsinothersituationsofsingu- laritytheory.Inthispaper,wefocusontwoofthem: (1)Letf:( R n+k ,0) −→ ( R k ,0)bearealanalyticgermwithan isolatedcriticalpointattheorigin.J.Milnor[11,Chapter11] provedthatforeverysufficientlysmallsphere S ε n+k − 1 centered attheoriginin R n+k ,thecomplement S ε n+k − 1 \ L f ofthelink L f = S ε n+k−1 ∩ f −1 (0)fibresoverthecircle.Aspointedoutby Milnor,thefibrationisnotnecessarilygivenbytheMilnormap f . kfk Thisresultcanbeextendedtoarealanalyticgermf: (X,p) −→ ( R k ,0)withisolatedcriticalvaluesatisfyingasuit- ablestratificationcondition,where(X,p)isagermofrealan- alyticspacewithisolatedsingularityatp([15],Theorem1.1). (2)Anotherdirectiondealswithlinksatinfinity.Letf: C n −→ C beapolynomialmap.Thelinkatinfinityassociatedwiththe fibref − 1 (0)isdefinedbyL f, ∞ =f − 1 (0) ∩ S 2 R n − 1 forasufficiently largeradiusR  1.In[12],A.Ne´methiandA.Zahariaproved Date:January29,2007. 1