J London Math Soc C London Mathematical Society doi:10 jlms jdp027
15 pages
English

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15 pages
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Niveau: Supérieur, Doctorat, Bac+8
J. London Math. Soc. (2) 80 (2009) 311–325 C!2009 London Mathematical Society doi:10.1112/jlms/jdp027 Milnor fibrations of meromorphic functions Arnaud Bodin, Anne Pichon and 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 punctured discs around the critical values of f/g, and the Milnor fibration on a sphere. 1. Introduction The classical fibration theorem of Milnor in [6] says that every holomorphic 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 ? = f |f | : S? \K ?? S1, (1) 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 f : B? ? f?1(∂D?) ?? ∂D? ?= S1, (2) 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 ?.

  • truncated global

  • milnor's proof concerns

  • fibre bundle

  • see also

  • global milnor

  • naturally associated

  • meromorphic germ

  • fibration over

  • milnor fibration


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Nombre de lectures 108
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J. London Math. Soc. (2) 80 (2009) 311–325 C ! 2009 London Mathematical Society doi:10.1112/jlms/jdp027
Milnor fibrations of meromorphic functions
Arnaud Bodin, Anne Pichon and Jos´e 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 punctured discs around the critical values of f /g , and the Milnor fibration on a sphere.
1. Introduction The classical fibration theorem of Milnor in [ 6 ] says that every holomorphic map (germ) f : ( C n , 0) ! ( C , 0) with n ! 2 and a critical point at 0 " C n has two naturally associated fibre bundles, and both of these are equivalent. The first is ! = | ff | : S ! \ K #! S 1 , (1) where S ! is a su ! ciently small sphere around 0 " C n and K = f ! 1 (0) $ S ! is the link of f at 0. The second fibration is f : B ! $ f ! 1 ( " D " ) #! " D " % = S 1 , (2) where B ! is the closed ball in C n with boundary S ! and D " is a disc around 0 " C , which is su ! ciently 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 di " eomorphic to S ! minus an open regular neighbourhood T of K . (Thus, to get the equivalence of the two fibrations one has to ‘extend’ the latter fibration to T \ K .) In fact, in order to have the second fibration one needs to know that every map-germ f as above has the so-called ‘Thom property’, which was not known when Milnor wrote his book. What he proved is that the fibres in ( 1 ) are di " eomorphic to the intersection f ! 1 ( t ) $ B ! for t close enough to 0. The statement that ( 2 ) is a fibre bundle was proved later in [ 5 ]byLˆeinthemoregeneralsettingof holomorphic maps defined on arbitrary complex analytic spaces, and we call it the MilnorLeˆ fibration of f . Once we know that ( 2 ) is a fibre bundle, the arguments of [ 6 , Chapter 5] show that this is equivalent to the Milnor fibration ( 1 ). The literature about these fibrations is vast, and so are their generalizations to various settings, including real analytic map-germs and meromorphic maps, and that is the starting point of this article. Let U be an open neighbourhood of 0 in C n and let f, g : U ! C be two holomorphic functions without common factors such that f (0) = g (0) = 0. We consider the meromorphic function F = f /g : U ! C P 1 defined by ( f /g )( x ) = [ f ( x ) /g ( x )]. As in [ 3 ], two such germs at 0, F = f /g and F " = f " /g " are considered as equal (or equivalent) if and only if f = hf " and g = hg " for some holomorphic germ h : C n ! C such that h (0) & = 0. Note that f /g is not defined on the
Received 7 April 2008; revised 20 January 2009; published online 15 June 2009. 2000 Mathematics Subject Classification 14J17, 32S25, 57M25. The research for this article was partially supported by the CIRM at Luminy, France, through a ‘Groupe de Travail’; there was also partial financial support from the Institut de Math´ematiques de Luminy and from CONACYT and PAPIIT-UNAM, Mexico.
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