OON THE MILNOR FIBRATION FOR f ( z )¯ g ( z ) II MUTSUO OKA
Abstract.
We consider a mixed function of type H ( z , ¯ z ) = f ( z )¯ g ( z )where f and g are holomorphic functions which are non-degenerate withrespect to the Newton boundaries. We assume also that the variety f = g = 0 is a non-degenerate complete intersection variety. In ourprevious paper, we considered the case that f, g are convenient so thatthey have isolated singularities. In this paper we do not assume theconvenience of f and g . In non-convenient case, two hypersurfaces mayhave non-isolated singularities at the origin. We will show that H hasstill both a tubular and a spherical Milnor fibrations under the localtame non-degeneracy and the toric multiplicity condition. We provealso the equivalence of two fibrations. Locally tame non-degenerate complete intersection pair
Introduction.
Let f ( z ) and g ( z ) be holomorphic functions vanishingat the origin. For h ( z ) := f ( z ) g ( z ), there exists a tubular Milnor fibration h : E ( r, δ ) ∗ → D ∗ δ or a spherical Milnor fibration h/ | h | : S r \ K r → S forsmall r and δ (cid:28) r ([15, 11]. Here E ( r, δ ) ∗ := { z ∈ B nr | (cid:54) = | f ( z ) ≤ δ } and K r := f − (0) ∩ S n − r . We consider the mixed function H ( z , ¯ z ) := f ( z )¯ g ( z )and the existence problem of its Milnor fibration. The link of H is the sameas the complex link given by h ( z ) but the fibration structure along the linkof g = 0 is conversely oriented. It turns out that such a fibration doesnot exist for an arbitrary pair. This problem has been studied by severalauthors but there are not yet satisfactory results ([26, 27, 28, 24]). For anon-degenerate mixed function, it is known that the Milnor fibration exists([19]). However for n ≥ H can not be non-degenerate as f = g = 0 is anon-isolated singular locus for H . In our previous paper [23], we have shownthe existence of Milnor fibrations for H under the assumption that f, g areconvenient non-degenerate functions, satisfying the multiplicity condition.A convenient non-degenerate function f has an isolated singularity at theorigin. In this paper, we consider the same problem without assuming theconvenience. That is, we consider the case that f = 0 or g = 0 may havenon-isolated singularity at the origin.1.2. Vanishing coordinate subspaces and locally tameness.
Let f ( z )be a holomorphic function of n complex variables z , . . . , z n which vanishes Mathematics Subject Classification.
Key words and phrases. locally tame, non-degenerate, toric multiplicity condition. a r X i v : . [ m a t h . AG ] J a n M. OKA at the origin. Consider a coordinate subspace C I := { ( z , . . . , z n ) ∈ C n | z j =0 , j / ∈ I } where I ⊂ { , , . . . , n } . C I is called a vanishing coordinate sub-space of f if the restriction of f to C I is identically zero. The restriction of f is denoted as f I . We denote the set of vanishing subspaces of f (respectivelyof g ) by V f (resp. by V g ). Let P = ( p , . . . , p n ) be a semi-positive weightvector. We put I ( P ) := { i | p i = 0 } . Take a vanishing coordinate subspace C I and take an arbitrary semi-positive weight vector P = ( p , . . . , p n ) suchthat I ( P ) = I . Then the face function f P is a weighted homogeneous func-tion of the variables ( z j ) j / ∈ I with a positive degree d ( P ; f ) with respect tothe weight vector P .Recall that f is non-degenerate if for any strictly positive weight vector P (i.e., I ( P ) = ∅ ), f P : C ∗ n → C has no critical points ([16]). We say that thefunction f (or the hypersurface V ( f ) := f − (0)) is locally tame and non-degenerate if it is non-degenerate and for any vanishing coordinate subspace C I , there exists a positive number r I such that for any weight vector P with I ( P ) = I , f P is a non-degenerate function of ( z j ) j / ∈ I with the othervariables ( z i ) i ∈ I ∈ C ∗ I being fixed in the ball (cid:80) i ∈ I | z i | ≤ r I ([21, 8]). Put V ( f ) (cid:93) := ∪ C I / ∈V f V ( f I ) ∩ C ∗ I . Recall that V ( f ) (cid:93) is smooth near the origin(Lemma (2.2), [17]).For the pair of function { f, g } , consider the following conditions.(1) The hypersurfaces V ( f ) = f − (0) , V ( g ) = g − (0) are locally tameand non-degenerate.(2) The variety V ( f, g ) = { f = g = 0 } is a locally tame non-degeneratecomplete intersection variety. Namely (2-a) for any strictly posi-tive weight vector P , the variety { z ∈ C ∗ n | f P ( z ) = g P ( z ) = 0 } is a smooth complete intersection variety. (2-b) For any commonvanishing coordinate subspace C I , there exists a positive number r I such that for any weight vector P with I ( P ) = I , { f P = g P = 0 } isa non-degenerate complete intersection variety in C ∗ J with J = I c and z I ∈ C ∗ I is fixed in the ball (cid:80) i ∈ I | z i | ≤ r I .We say { f, g } is a locally tame non-degenerate pair if it satisfies only (1) and(2). The pair { f, g } is a disjoint locally tame non-degenerate complete in-tersection pair if it satisfies (1), (2-a) and (3) { f, g } satisfies the disjointnessof vanishing subspaces, i.e. V f ∩ V g = ∅ .Note that if C I ∈ V f \ V g , there exists a positive number r I such that forany semi-positive weight vector P with I ( P ) = I , g P = g I and f P = g P = 0is a non-degenerate complete intersection variety. Remark 1. If f is locally tame and non-degenerate and if C I is not a van-ishing coordinate subspace for f , f I is also locally tame and non-degenerateas a function on C I . See the argument in Proposition (1.5), Chapter III [17] . Locally tameness has been defined for mixed functions (Definition 2.7, [8] ). If a holomorphic function f ( z ) is locally tame, it is also locally tameas a mixed function. N THE MILNOR FIBRATION FOR f ( z )¯ g ( z ) II 3 In this paper, we consider the existence problem of the Milnor fibrationof H ( z , ¯ z ) = f ( z )¯ g ( z ) under the assumption that f, g is a locally tamenon-degenerate complete intersection pair. For the further detail about anon-degenerate complete intersection variety, see [16, 17].1.3. Examples.
Let f ( z ) = z a + · · · + z an − + z n − z a − n with a >
1. Then C I , I = { n } is a vanishing coordinate subspace for f . Let g ( z ) = c z a − z + c z a + · · · + c n z an and g = c z a + · · · + c n − z an − + c n z n − z an . For genericcoefficients c , . . . , c n , f, g , g are locally tame non-degenerate functions and V f = V g = { C I } and V g = { C J } where I = { n } and J = { } . { f, g } isa disjoint locally tame non-degenerate pair while { f, g } is a locally tamenon-degenerate pair. They have the common vanishing coordinate subspace C { n } . 2. Isolatedness of the critical value
Multiplicity condition.
We slightly generalize the multiplicity condi-tion which is introduced in [23]. We say that H := f ¯ g satisfies the multiplic-ity condition if there exists a good resolution π : X → C n of the holomorphicfunction h := f g such that(i) π : X \ π − ( V ( h )) → C n \ V ( h ) is biholomorphic and the divisordefined by π ∗ ( f g ) = 0 has only normal crossing singularities andthe respective strict transforms ˜ V ( f ) of V ( f ) and ˜ V ( g ) of V ( g ) aresmooth.(ii) Put π − ( ) = ∪ sj =1 D j where D , . . . , D s are smooth compact divi-sors in X . Denote the respective multiplicities of π ∗ f and π ∗ g along D j by m j and n j . Then m j (cid:54) = n j for j = 1 , . . . , s .Assume that there exists a regular simplicial cone subdivision Σ ∗ of the dualNewton diagram Γ ∗ ( f g ) and let ˆ π : X → C n be the corresponding admissibletoric modification. Let V + be the set of strictly positive vertices of Σ ∗ . Thenit gives a good resolution of the function f g and the compact exceptionaldivisors are bijectively correspond to { ˆ E ( P ) | P ∈ V + } ([17]). Recall that themultiplicity of ˆ π ∗ f and ˆ π ∗ g along the divisor ˆ E ( P ) are given by d ( P, f ) and d ( P, g ) respectively. We say that ˆ π satisfies the toric multiplicity conditionfor H if d ( P, f ) (cid:54) = d ( P, g ) , ∀ P ∈ V + . For further detail about the toric modification ˆ π : X → C n , we refer to [17]. Lemma 2 ( Isolatedness of the critical value, Lemma 3 [23]) . Assume that { f, g } is a locally tame and non-degenerate complete intersection pair. As-sume that there exists an admissible toric modification ˆ π : X → C n whichsatisfies the toric multiplicity condition. Then there exist positive numbers r such that is the unique critical value of H on B nr . The proof follows by the exact same argument as Lemma 3,[23].
M. OKA
A sufficient condition for the toric multiplicity condition.
We considerthe following truncated cone. Let h ( z ) = (cid:80) ν a ν z ν be a holomorphic functionwhich is not necessarily convenient. Let Γ + ( h ) be the convex hull of theunion (cid:83) ν,a ν (cid:54) =0 { ν + ( R + ) n } as usual. The Newton boundary Γ( h ) is definedby the union of compact faces of Γ + ( h ). To give a sufficient condition forthe multiplicity condition, we further consider following. Definition 3.
We define the set Γ ++ ( h ) and IntΓ ++ (h) as Γ ++ ( h ) = { rν | r ≥ , ν ∈ Γ( h ) } , IntΓ ++ (h) = { r ν | r > , ν ∈ Γ(h) } Note that Γ ++ ( h ) ⊂ Γ + ( h ) and the equality holds if and only if h isconvenient. The following gives a sufficient condition for the multiplicitycondition. Lemma 4.
Assume { f, g } is a locally tame non-degenerate complete inter-section pair. Suppose the following condition is satisfied. ( (cid:93) ) : Γ( f ) ⊂ Int Γ ++ (g) or Γ( g ) ⊂ Int Γ ++ (f) .Then the multiplicity condition is satisfied with respect to any admissibletoric modification. See Figure 1 which shows the situation Int Γ ++ (f) ⊃ Γ(g). The con-dition ( (cid:93) ) is a generalization of Newton multiplicity condition in [23] fornon-convenient f and g . We call ( (cid:93) ) the tame Newton multiplicity condition . Example 5.
1. Assume that f ( z ) (respectively g ) is a convenient functionand assume that Γ( f ) ∩ Γ( g ) = ∅ and Γ( g ) is above Γ( f ) (resp. Γ( f ) is above Γ( g ) ). Then the tame Newton multiplicity condition is satisfied.2. Assume { f, g } is a locally tame non-degenerate complete intersectionpair and let ˆ π : X → C n is an admissible toric modification. Let V + be thestrictly positive vertices of Σ ∗ . Consider the mapping ϕ m : C n → C n definedby ϕ ( z ) = ( z m , . . . , z mn ) and put f m ( z ) := ϕ ∗ f ( z ) and g m ( z ) := ϕ ∗ g ( z ) .Then there exists a sufficiently large m such that ˆ π : X → C n satisfies thetoric multiplicity condition for f m ¯ g and f ¯ g m respectively. This follows fromthe canonical equality d ( P, f m ) = m d ( P, f ) and d ( P, g m ) = m d ( P, g ) andthe stability of the dual Newton diagrams Γ ∗ ( f ) = Γ ∗ ( f m ) , Γ ∗ ( g ) = Γ ∗ ( g m ) .3. Let f ( z ) = c z a + · · · + c n z an . Let g ( z ) be any locally tame non-degenerate function. Then { f, g } is a locally tame non-degenerate pair forgeneric coefficients c , . . . , c n and satisfies the Newton multiplicity conditionif a = 1 . If a > , { f, g (cid:81) ni =1 z ai } satisfies the Newton multiplicity condition,as Γ ++ ( f ) ⊃ IntΓ ++ (g) .Alternative proof of Lemma 2. Though Lemma 2 follows from Lemma 3,[23] under the assumption of the tame Newton multiplicity condition ( (cid:93) ),it will be useful to give another proof which does not use a resolution. Weprove the assertion by contradiction. Assume that the assertion does nothold. Then using the Curve Selection Lemma ([15, 10]), we can find ananalytic path z ( t ) , ≤ t ≤ z (0) = such that H ( z ( t ) , ¯ z ( t )) (cid:54) = 0 for t (cid:54) = 0 N THE MILNOR FIBRATION FOR f ( z )¯ g ( z ) II 5 Γ (f) Γ (g) Γ ++ ( g )Γ ++ ( f ) Figure 1. Γ( g ) ⊂ Int Γ ++ (f)and z ( t ) is a critical point of the function H : C n → C for any t . UsingProposition 1, [18] (see also Proposition 1, [23]), we can find an analyticfunction λ ( t ) whose values are in S ⊂ C such that ∂H ( z ( t ) , ¯ z ( t )) = λ ( t ) ¯ ∂H ( z ( t ) , ¯ z ( t )) . (1)Note that in our case we have ∂H = ∂f g, ¯ ∂H = f ∂g. Thus (2) implies g ( z ( t )) ∂f ( z ( t )) = λ ( t ) f ( z ( t )) ∂g ( z ( t )) . (2)Put I = { j | z j ( t ) (cid:54)≡ } . We may assume for simplicity that I = { , . . . , m } and we consider the restriction H I = H | C I . By the assumption z ( t ) / ∈ V ( H )for t (cid:54) = 0, we see that H I (cid:54) = 0. Consider the Taylor expansions of z ( t ) and λ ( t ): z i ( t ) = b i t p i + (higher terms) , b i (cid:54) = 0 , p i > , i = 1 , . . . , mλ ( t ) = λ + (higher terms) , λ ∈ S ⊂ C . Consider the weight vector P = ( p , . . . , p m ) and the point b = ( b , . . . , b m ) ∈ C ∗ I and also the face function f IP of f I ( z , ¯ z ). Recall that f IP and g IP are de-fined by the partial sum of monomials in f I ( z , ¯ z ) where the monomials have M. OKA the minimal degree d ( P ; f I ) and d ( P ; g I ) respectively. Then we have f ( z ( t )) = f IP ( b ) t d ( P ; f I ) + (higher terms) ,g ( z ( t )) = g IP ( b ) t d ( P ; g I ) + (higher terms) ,∂f∂z j ( z ( t )) = ∂f IP ∂z j ( b ) t d ( P ; f ) − p j + (higher terms) , j ∈ I∂g∂z j ( z ( t )) = ∂g IP ∂z j ( b ) t d ( P ; g ) − p j + (higher terms) , j ∈ I. The equality (2) says that ∂f I ∂z j ( z ( t ) , ¯ z ( t )) = λ ( t ) f ( z ( t )) g ( z ( t )) ∂g I ∂z j ( z ( t ) , ¯ z ( t )) , j = 1 , . . . , m. (3)Put (cid:96) := ord f ( z ( t )) − ord g ( z ( t )) = ord t f ( z ( t )) g ( z ( t )) . Here ord ϕ ( t ) of a Lau-rent series ϕ ( t ) is by definition the lowest degree of the series ϕ ( t ). Thuslim t → ϕ ( t ) /t ord ϕ ( t ) is a non-zero number. Note thatord f ( z ( t )) ≥ d ( P ; f ) , ord g ( z ( t )) ≥ d ( P ; g ) , ord ∂f∂z j ( z ( t )) ≥ d ( P ; f ) − p j , ord ∂g∂z j ( z ( t )) ≥ d ( P ; g ) − p j . Case 1. (a) f IP ( b ) = 0 , g IP ( b ) (cid:54) = 0 or (b) f IP ( b ) (cid:54) = 0 , g IP ( b ) = 0. In thecase of (a), (cid:96) > d ( P ; f I ) − d ( P ; g I ) and (8) says that ∂f IP ( b ) = 0. Thisimplies b is a critical point of f IP and a contradiction to the non-degeneracyassumption. In case of (b), (cid:96) < d ( P ; f I ) − d ( P ; g I ) we get ∂g IP ( b ) = 0 andwe get also a contradiction to the non-degeneracy assumption of g .Case 2. f P ( b ) = 0 , g P ( b ) = 0. Then b ∈ V ( f P , g P ). If (cid:96) (cid:54) = d ( P ; f ) − d ( P ; g ), we get a contradiction ∂f IP ( b ) = 0 or ∂g IP ( b ) = 0. So we assumethat (cid:96) = d ( P ; f ) − d ( P ; g ). Then we see that ∂f IP ( b ) , ∂g IP ( b ) are linearly de-pendent over C but this is a contradiction to the non-degeneracy assumptionof the intersection variety V ( f, g ).Case 3. Assume that f P ( b ) (cid:54) = 0 , g P ( b ) (cid:54) = 0. Then we have (cid:96) = d ( P ; f ) − d ( P ; g ) and g P ( b ) ∂f P ∂z j ( b ) = λ f P ( b ) ∂g P ∂z j ( b ) , j = 1 , . . . , m. Multiplying p j ¯ b j to the both side and summing up for j = 1 , . . . , m , we usethe Euler equalities of f P and g P , d ( P, f I ) f IP ( b ) = m (cid:88) j =1 p j b j ∂f IP ∂z j ( b ) , d ( P, g I ) g IP ( b ) = m (cid:88) j =1 p j b j ∂g IP ∂z j ( b ) , to get the equality d ( P ; f ) f P ( b ) g P ( b ) = λ d ( P ; g ) f P ( b ) g P ( b ) or N THE MILNOR FIBRATION FOR f ( z )¯ g ( z ) II 7 This gives an absurd equality:1 (cid:54) = d ( P ; f ) d ( P ; g ) = (cid:12)(cid:12)(cid:12)(cid:12) λ f P ( b ) g P ( b ) f P ( b ) g P ( b ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 . The first inequality follows from the tame Newton multiplicity condition.The last equality is due to | λ | = 1. (cid:3) Remark 6. If f is locally tame and non-degenerate and if C I is not a van-ishing coordinate subspace for f , f I is also locally tame and non-degenerateas a function on C I . See the argument in Proposition (1.5), Chapter III [17] . Locally tameness has been defined for mixed functions (Definition 2.7, [8] ). If a holomorphic function f ( z ) is locally tame, it is also locally tameas a mixed function. Fibration problem for function f ¯ g We study the existence problem for the Milnor fibration of the mixedfunction H ( z , ¯ z ) := f ( z )¯ g ( z ) in a more general situation. In this paper,we do not assume the convenience of f and g and therefore V ( f ) or V ( g )may have non-isolated singularities at the origin. There are also interestingworks from more general viewpoint in Parameswaran and Tibar [25, 24] andAraujo dos Santos, Ribeiro and Tibar [5] where authors consider the case ofcritical values being not isolated.3.1. Canonical stratification.
We assume that { f, g } is a locally tamenon-degenerate complete intersection pair. Consider the hypersurface V ( f g ) = V ( f ) ∪ V ( g ). Note that the mixed hypersurface V ( f ¯ g ) is equal to V ( f g ) asreal algebraic varieties. We consider the following canonical stratification S of C ∗ n which also give a stratification of V ( f g ). Put V ∗ I ( f ) = V ( f I ) ∩ C ∗ I if C I is not a vanishing coordinate subspace.Here C ∗ I = { ( z i ) ∈ C I | ∀ z i (cid:54) = 0 , i ∈ I } . We first define a stratification S I of C ∗ I as follows. (cid:8) C ∗ I \ ( V ∗ I ( f ) ∪ V ∗ I ( g )) , V ∗ I ( f ) (cid:48) , V ∗ I ( g ) (cid:48) , V ∗ I ( f ) ∩ V ∗ I ( g ) (cid:9) , if f I (cid:54) = 0 , g I (cid:54) = 0 (cid:8) C ∗ I \ V ∗ I ( f ) , V ∗ I ( f ) (cid:9) , if g I ≡ , f I (cid:54) = 0 (cid:8) C ∗ I \ V ∗ I ( g ) , V ∗ I ( g ) (cid:9) , if f I ≡ , g I (cid:54) = 0 (cid:8) C ∗ I (cid:9) , if f I ≡ , g I ≡ . and we define S = ∪ I S I . If { f, g } is a disjoint locally tame non-degeneratepair, the last case does not exist. Here V ∗ I ( f ) (cid:48) = V ∗ I ( f ) \ V ∗ I ( g ) and V ∗ I ( g ) (cid:48) = V ∗ I ( g ) \ V ∗ I ( f ). V I ( f ) is empty only if f I is a monomial.We call S the canonical toric strafitication of V ( f g ) = V ( f ¯ g ). Note that S is a complex analytic stratification. M. OKA
Transversality and Thom’s a f -regularity. We use the notation V ( H, z ) := H − ( H ( z )) hereafter. Another key condition for the existence ofthe Milnor fibration is the transversality of the nearby fibers H − ( η ) , η (cid:54) = 0and the sphere S n − r . Assume that 0 is the unique critical value of H in B nr . Transversality of nearby fibers : For any pair r ≤ r , there exists apositive number δ such that for any r, r ≤ r ≤ r and non-zero η with | η | ≤ δ , H − ( η ) and S n − r intersect transversely. This condition followsif H satisfies the Thom’s a f -regularity (See for example, Proposition 11,[21]). Recall that H satisfies a f -condition at the origin if there exists astratification S of H − (0) ∩ B nr for some r > q ν , ν = 1 , , . . . which converges q ∈ M, M ∈ S and q (cid:54) = , the limit ofthe tangent space T q ν V ( H, q ν ) (if it exists) includes the tangent space of M at q . Theorem 7.
Assume that either (i) { f, g } is a locally tame non-degeneratecomplete intersection pair which satisfies also the tame Newton multiplicitycondition ( (cid:93) ) or (ii) { f, g } is a disjoint locally tame non-degenerate completeintersection pair. In the case (ii), we assume also that H has a uniquecritical value in a ball B nr . Then H = f ¯ g satisfies a f -regularity. Note that in case (i), the tame Newton multiplicity condition guaranteesthe isolatedness of the critical value of H . For the proof, we consider thecanonical toric stratification S on V ( f g ). We choose r , r ≥ r > r ≤ r , the canonical toric strata are smoothin B nr and any sphere S n − ρ with 0 < ρ ≤ r meets transversally withevery strata of S of positive dimension. We use Curve selection lemma (see[15, 10]). Suppose we have a real analytic curve z ( t ) , ≤ t ≤ z (0) = a ∈ V ( H ) ∩ B nr , a (cid:54) = and z ( t ) ∈ C n \ V ( H ) for t >
0. Put K = { i | z i ( t ) (cid:54)≡ } and write the expansion as z i ( t ) = α i t p i + (higher terms) , α i (cid:54) = 0 , i ∈ K ≡ , i / ∈ K Let M ∈ S I be the stratum which contains a . We have to show that thelimit of the tangent space of the fiber V ( H, z ( t )) at z ( t ) for t → M at a . The restriction of f, g and H on C K satisfy also the locally tame non-degenerate assumption. As theargument for the proof is exactly the same, we assume for simplicity that K = { , . . . , n } hereafter. That is, we assume that z ( t ) ∈ C ∗ n for t (cid:54) = 0 and z (0) = a . Put P = ( p , . . . , p n ) and (cid:40) I := { i | p i = 0 } ,J = I c = { , . . . , n } \ I, w := ( α , . . . , α n ) ∈ C ∗ n . (4) N THE MILNOR FIBRATION FOR f ( z )¯ g ( z ) II 9 Note that p i = 0 if and only if i ∈ I . Thus a = w I and 0 (cid:54) = (cid:107) a (cid:107) ≤ r . Wewill show that lim t → T z ( t ) V ( H, z ( t )) ⊃ T a M. We use the key property that the tangent space of the level hypersurface V ( H, z ( t )) at z ( t ) contains the intersection of two tangent spaces of thelevel complex hypersurfaces V ( f, z ( t )) and V ( g, z ( t )) by Proposition 14, [23].Here we are assuming that r is sufficiently small so that 0 is the only criticalvalue for f and g on B nr . We divide the situation into three cases.(a) C I / ∈ V f ∪ V g , i.e. f I (cid:54)≡ , g I (cid:54)≡ C I ∈ V f and C I / ∈ V g , i.e. f I ≡ , g I (cid:54)≡ b ) (cid:48) C I ∈ V g and C I / ∈ V f , i.e. f I (cid:54)≡ g I ≡ C I ∈ V f ∩ V g , i.e. f I ≡ g I ≡ b ) (cid:48) is symmetric, it is enough to consider three cases (a), (b)and (c).We first consider the case (a). The case (a) can be divided into twosubcases:(a-1) a ∈ M = V ∗ I ( f ) ∩ V ∗ I ( g ).(a-2) a ∈ V ∗ I ( f ) (cid:48) = V ∗ I ( f ) \ V ∗ I ( g ), or a ∈ V ∗ I ( g ) (cid:48) = V ∗ I ( g ) \ V ∗ I ( f ).In the case (a-1), a is a non-singular point of a ∈ V ( f, g ). As the tan-gent space T z ( t ) V ( f, z ( t )) converges to T a V ( f ) which includes T a V ∗ I ( f ) and T z ( t ) V ( g, z ( t ))) converges to T a V ( g ) which includes T a V ∗ I ( g ) and T a M = T a V ∗ I ( f ) ∩ T a V ∗ I ( g ) by the Newton non-degeneracy assumption, the asser-tion follows from Proposition 14, [23].In the case (a-2), a ∈ V ∗ I ( f ) (cid:48) or a ∈ V ∗ I ( g ) (cid:48) , a is a non-singular point of V ( H ) and the assertion is obvious from the continuity of the tangent space.Consider the case (b). Thus we assume that C I ∈ V f \ V g . By thelocal tameness assumption, the limit of the normalized holomorphic gradientvector lim t → ∂ f ( z ( t )) / (cid:107) ∂ f ( z ( t )) (cid:107) along z ( t ) is a vector in C J . Here J = { , . . . , n } \ I . (Recall ∂f ( z ) = ( ∂f ( z ) ∂z , . . . , ∂f ( z ) ∂z n ).) Thus the limit of thetangent space of V ( f, z ( t )) contains C I by the local tameness assumption.There are two subcases.(b-1) a ∈ V ∗ I ( g ), or(b-2) a ∈ C ∗ I \ V ∗ I ( g ).Note that M = V ∗ I ( g ) in the case (b-1) and M = C ∗ I \ V ∗ I ( g ) in the case(b-2) respectively. In the case of (b-1), the limit of the normalized vectorof ∂f ( z ( t )) is a vector in C J by the local tameness assumption of f . Thusthe limit of T z ( t ) V ( f, z ( t )) includes C I . On the other hand, as ∂g I ( a ) isnon-zero, T a V ( g ) is transverse to C I at a . Thus for any sufficiently small t ,they are transverse and the limit of the intersection of two tangent space ofthe tangent space of V ( f, z ( t )) and V ( g, z ( t )) contains T a V ∗ I ( g ). Now we consider the case (b-2). We claim that the limit of the tangentspace T z ( t ) V ( H, z ( t )) includes C I , the tangent space of the stratum M = C ∗ I \ V ∗ I ( g ) at a . First we prepare a sublemma. Sublemma 8.
Let f be a holomorphic function and write f ( z ) = k ( z , ¯ z ) + i(cid:96) ( z , ¯ z ) where k = (cid:60) f, (cid:96) = (cid:61) f . Then we have ¯ ∂k = ∂f and ¯ ∂(cid:96) = i ∂f .In particular, two gradient vectors ¯ ∂k and ¯ ∂(cid:96) are linearly dependent over C but linearly independent over R at a non-critical point z of f . The assertion follows from the identities: ∂k = ¯ ∂k, ∂(cid:96) = ¯ ∂(cid:96), ¯ ∂f = ¯ ∂k + i ¯ ∂(cid:96) = 0 , ∂f = ∂k + i∂(cid:96). Put p min = min { p j | j / ∈ I } . First we can write Lemma 9.
The orders of ¯ ∂ (cid:60) f ( z ( t )) and ¯ ∂ (cid:61) f ( z ( t )) are equal to the orderof ∂f ( z ( t )) . Put s = order ∂f ( z ( t )) . Then s and strictly less than d ( P ; f ) − p min . We can write further as follows. ∂f ( z ( t )) = v t s + (higher terms) , ∃ v ∈ C J ¯ ∂ (cid:60) f ( z ( t )) = 12 v t s + (higher terms) ¯ ∂ (cid:61) f ( z ( t )) = i v t s + (higher terms)In particular, lim t → T z ( t ) V ( f, z ( t )) is the complex orthogonal of v . Now we are ready to analyze the case (b-2). Note that the limit of nor-malized gradient vector ∂f ( z ( t )) is v / (cid:107) v (cid:107) . For a vector v , let v ⊥ C be thesubspace of C n which are complex orthogonal to v . Namely v ⊥ C = { w ∈ C n | ( w , v ) = 0 } . Now we claim Assertion 10.
Assume a ∈ C ∗ I \ V ∗ I ( g ) . Then lim t → T z ( t ) V ( H, z ( t )) =lim t → T z ( t ) V ( f, z ( t )) .Proof. Put b := ¯ g ( a ) and write b = b + ib with b , b ∈ R . First we use theequalities: (cid:60) ( H ) = (cid:60) ( f ) (cid:60) (¯ g ) − (cid:61) ( f ) (cid:61) (¯ g ) , (cid:61) ( H ) = (cid:60) ( f ) (cid:61) (¯ g ) + (cid:61) ( f ) (cid:60) (¯ g ) . N THE MILNOR FIBRATION FOR f ( z )¯ g ( z ) II 11 Then the gradient vectors are given as¯ ∂ (cid:60) ( H )( z ( t )) = ( ¯ ∂ (cid:60) ( f ) (cid:60) (¯ g )( z ( t )) + ( (cid:60) ( f ) ¯ ∂ (cid:60) (¯ g ))( z ( t )) − ( ¯ ∂ (cid:61) ( f ) (cid:61) (¯ g ))( z ( t )) − ( (cid:61) ( f ) ¯ ∂ (cid:61) (¯ g ))( z ( t )) ≡ b ¯ ∂ (cid:60) ( f )( z ( t )) − b ¯ ∂ (cid:61) f ( z ( t )) modulo (t s+1 ) ≡ v ¯ b t s modulo (t s+1 )¯ ∂ (cid:61) ( H )( z ( t )) = ( ¯ ∂ (cid:60) ( f ) (cid:61) ¯ g )( z ( t )) + ( (cid:60) ( f ) ¯ ∂ (cid:61) (¯ g ))( z ( t ))+ ¯ ∂ (cid:61) ( f ) (cid:60) (¯ g )( z ( t )) + (cid:61) ( f ) ¯ ∂ (cid:60) (¯ g )( z ( t )) ≡ b ( ¯ ∂ (cid:60) ( f )( z ( t )) + b ¯ ∂ (cid:61) ( f )( z ( t )) modulo (t s+1 ) ≡ ( i v ¯ b )2 t s modulo (t s+1 )and therefore the normalized vector of these gradient vectors ¯ ∂ (cid:60) ( H )( z ( t ))and ¯ ∂ (cid:61) ( H )( z ( t )) converges to the vectors v ¯ b (cid:107) v ¯ b (cid:107) , i v ¯ b (cid:107) v ¯ b (cid:107) respectively. This implies the limit of the tangent space T z ( t ) V ( H, z ( t )) isthe real orthogonal of the real 2-dimensional subspace span by these twovectors, that is nothing but the complex subspace v ⊥ C which is equal to thelimit of T z ( t ) V ( f, z ( t )). The proof of the assertion for (b-2) is now completed.The case { f, g } is a disjoint tame non-degenerate complete intersection pairis now proved.Now we consider the last case (c) C I ∈ V f ∩ V g . Recall that w =( α , . . . , α n ). We divide the situation into three subcases.(c-1) f P ( w ) = g P ( w ) = 0.(c-2) f P ( w ) = 0 and g P ( w ) (cid:54) = 0 or (c-2)’ f P ( w ) (cid:54) = 0 and g P ( w ) = 0.(c-3) f P ( w ) (cid:54) = 0 , g P ( w ) (cid:54) = 0.We restate the assertion as the following lemma. Lemma 11.
Assume that C I ∈ V f ∩ V g . The the limit of the tangent space T z ( t ) V ( H, z ( t )) includes C I as a subspace.Proof. First assume that f P ( w ) = g P ( w ) = 0. Put ∂f ( z ( t )) = ( u ( t ) , . . . , u n ( t ))and ∂g ( z ( t )) = ( v ( t ) , . . . , v n ( t )). We can write as u j ( t ) = ∂f P ∂z j ( w ) t d ( P ; f ) − p j + (higher terms) v j ( t ) = ∂g P ∂z j ( w ) t d ( P ; g ) − p j + (higter terms) . Put o f and o g be the orders of ∂f ( z ( t )) and ∂g ( z ( t )) respectively. Thatis o f = min { ord t u i ( t ) | i = 1 , . . . , n } and o g = min { ord v i ( t ) | i = 1 , . . . , n } . Then the limit of ∂f ( z ( t )) and ∂g ( z ( t )) up to scalar multiplications arerepresented respectively bylim t → t o f ∂f ( z ( t )) , lim t → t o g ∂g ( z ( t )) . (5)We denote these limit vectors as lim ( n ) t → ∂f ( z ( t )) and lim ( n ) t → ∂g ( z ( t )). Ifthese two limits are linearly independent over C , the intersection T z ( t ) V ( f, z ( t )) ∩ T z ( t ) V ( g, z ( t ))converges to the the complex orthogonal subspace to these two limit vectors.That is, < ∂f ( z ( t )) , ∂g ( z ( t )) > ⊥ C (cid:55)→ < lim t → n ) ∂f ( z ( t )) , lim t → n ) ∂g ( z ( t )) > ⊥ C . The problem happens if these two limits are linearly dependent. We use asimilar argument as the one which is used in the proof of Theorem 20, [21] orTheorem 3.14, [8] to solve this problem. For the simplicity of the argument,we assume that J = { , . . . , m } and I = { m + 1 , . . . , n } and we assume that p ≥ p ≥ · · · ≥ p m > , p m +1 = · · · = p n = 0 . Note that p min = p m under the above assumption andord u j ( t ) ≥ d ( P ; f ) − p j , ord v j ( t ) ≥ d ( P ; g ) − p j , j = 1 , . . . , m while for j ≥ m + 1,ord u j t ) ≥ d ( P ; f ) , ord v j ( t ) ≥ d ( P ; g ) , j = m + 1 , . . . , n. Now we consider first (c-1): f P ( w ) = g P ( w ) = 0. By the locally tamenon-degeneracy assumption, there exists 1 ≤ a, b ≤ m, a (cid:54) = b so that wehave det (cid:32) ∂f P ∂z a ( w ) ∂f P ∂z b ( w ) ∂g P ∂z a ( w ) ∂g P ∂z b ( w ) (cid:33) (cid:54) = 0 . (6)Here we assume that a (cid:54) = b but we do not assume that a < b . In particular, o f ≤ max { d ( P ; f ) − p a , d ( P ; f ) − p b } ≤ d ( P ; f ) − p m ,o g ≤ max { d ( P ; g ) − p a , d ( P ; g ) − p b } ≤ d ( P ; g ) − p m . For simplicity, we may assume that o f ≤ o g and consider (cid:96) := min { j | ord u j ( t ) = o f } , m := min { j | ord v j ( t ) = o g } . We call (cid:96) , m the leading indices of ∂f ( z ( t ) and ∂g ( z ( t )) . Case 1. Assume that (cid:96) (cid:54) = m . Then the two limit gradient vectors givenby (5) are already linearly independent. There are nothing to do further.Case 2. Assume that (cid:96) = m . Then we take a monomial function ρ ( t ) = ct o g − o f , c ∈ C and replace ∂g ( z ( t )) by v (1) ( t ) = ∂g ( z ( t )) − ρ ( t ) ∂f ( z ( t )) . N THE MILNOR FIBRATION FOR f ( z )¯ g ( z ) II 13 We choose a constant c so that ord v (1) m ( t ) > o g . We put ord v (1) m ( t ) = ∞ if v (1) m ( t ) ≡
0. Here v (1) m ( t ) is the m -th component of v (1) ( t ). Note that thetwo dimensional complex subspace W = (cid:10) ∂f ( z ( t )) , ∂g ( z ( t )) (cid:11) generated by { ∂f ( z ( t )) , ∂g ( z ( t )) } is the same with subspace (cid:10) ∂f ( z ( t )) , v (1) ( t ) (cid:11) generatedby { ∂f ( z ( t )) , v (1) ( t ) } . Thus their complex orthogonal subspaces are alsoequal. We continue this operation ∂g → v (1) → · · · → v ( k ) until the leading index of v ( k ) changes. Note that the k-times operation ∂g ( z ( t )) → v ( k ) ( t ) is given as v ( k ) ( t ) = ∂g ( z ( t )) − ρ k ( t ) ∂f ( z ( t ))where ρ k ( t ) is a polynomial of variable t whose lowest degree is o g − o f . By(6), we may assume that ∂g P /∂z a ( w ) (cid:54) = 0. Note that ord v ( ν ) m ( t ) is strictlyincreasing as long as ν ≤ k − v ( ν ) ( t ) = ord v ( ν ) m ( t ). Let us look thecomponents a, b which is given by v ( k ) τ ( t ) = ∂g∂z τ ( z ( t )) − ρ k ( t ) ∂f∂z τ ( z ( t )) , τ = a, b. Assertion 12.
One of the following inequalities holds. ord v ( k ) a ( t ) ≤ d ( P ; g ) − p a , or ord v ( k ) b ( t ) ≤ d ( P ; g ) − p b . Proof.
Assume that ord v ( k ) a > d ( P ; g ) − p a . We will show that ord v ( k ) b ( t ) ≤ d ( P ; g ) − p b . As the first term of ρ k ( t ) ∂f∂z a ( z ( t )) kill the first term of ∂g∂z a ( t ),the order of ρ k ( t ) ∂f∂z a ( t ) is equal to d ( P ; g ) − p a . There are two cases to beconsidered.(A) ∂f P ∂z a ( w ) (cid:54) = 0 or (B) ∂f P ∂z a ( w ) = 0.Assume the case (A). Then we have ord ρ k ( t ) = d ( P ; g ) − d ( P ; f ) = o g − o f and which implies thatord ρ k ( t ) ∂f P ∂z b ( z ( t )) ≥ ( d ( P ; f ) − p b ) + ( d ( P ; g ) − d ( P ; f )) = d ( P ; g ) − p b . Then putting λ be the coefficient of t d ( P ; g ) − d ( P ; f ) in ρ k ( t ), we have0 (cid:54) = det (cid:32) ∂f P ∂z a ( w ) ∂f P ∂z b ( w ) ∂g P ∂z a ( w ) ∂g P ∂z b ( w ) (cid:33) = det (cid:32) ∂f P ∂z a ( w ) ∂f P ∂z b ( w ) ∂g P ∂z a ( w ) + λ ∂f P ∂z a ( w ) ∂g P ∂z b ( w ) + λ ∂f P ∂z b ( w ) (cid:33) = det (cid:32) ∂f P ∂z a ( w ) ∂f P ∂z b ( w )0 ∂g P ∂z b ( w ) + λ ∂f P ∂z b ( w ) (cid:33) and thus ∂g P ∂z b ( w ) + λ ∂f P ∂z b ( w ) (cid:54) = 0 by (6), ord v ( k ) b = d ( P ; g ) − p b .Consider the case (B) now. Then ord ρ k ( t ) < d ( P ; g ) − d ( P ; f ). By (6), ∂f P ∂z b ( w ) (cid:54) = 0 and ord v ( k ) ( t ) b < d ( P ; g ) − p b . Thus in both cases, under theabove operation, ord v ( k ) ( t ) ≤ d − p m . (cid:3) The above argument implies that the number k of operations is boundedby k ≤ d ( P ; g ) − p m − o g . At the last operation, the leading index of v ( k ) ( t )is different from m and the limit vector of v ( k ) ( t ) and ∂f ( z ( t )) are linearlyindependent and they are in the subspace C J . As T z ( t ) V ( f, z ( t )) ∩ T z ( t ) V ( g, z ( t )) is the complex orthonormal subspace of thetwo dimensional subspace (cid:10) ∂f ( z ( t )) , ∂g ( z ( t )) (cid:11) C and it is equal to the com-plex orthonormal subspace of (cid:10) ∂f ( z ( t )) , v ( k ) ( t ) (cid:11) C , the limit of T z ( t ) V ( f, z ( t )) ∩ T z ( t ) V ( g, z ( t )) includes the vanishing subspace C I . As T z ( t ) V ( H, z ( t )) in-cludes T z ( t ) V ( f, z ( t )) ∩ T z ( t ) V ( g, z ( t )) as a subspace, lim t → T z ( t ) V ( H, z ( t )) ⊃ C I . Thus the proof of case (c-1) is done.Now we consider the case (c-2): f P ( w ) = 0 , g P ( w ) (cid:54) = 0. Consider the hy-persurface V := { ( z J ∈ C ∗ J | f P ( z ) = 0 , z I = a } . By the locally tamenessassumption, V is a non-singular hypersurface in C ∗ I . Let us consider therestriction g P : V → C . As g P is a weighted homogeneous polynomial func-tion of weight P J = ( p , . . . , p m ) and V is invariant under the associated C ∗ action on C J , g P has no non-zero critical value on V . Namely ¯ ∂g P ( w )is non-zero and linearly independent with ¯ ∂f P ( w ). Thus there is a pair a ≤ b ≤ m which satisfies (6). Thus the rest of the argument is the exactsame as above and lim s → T z ( s ) V ( H, z ( s )) ⊃ C I . The case (c-2)’: f P ( w ) (cid:54) = 0and g P ( w ) = 0 is treated similarly.Now we consider the case (c-3): f P ( w ) , g P ( w ) (cid:54) = 0. In this case, we needthe assumption that f, g satisfies the tame Newton multiplicity condition.Let deg P f = d f and deg P g = d g . Put d r := d f + d g and d p = d f − d g .Then the tame Newton multiplicity condition implies d p (cid:54) = 0. The mixedfunction H P = f P ¯ g P : C J → C is a strongly mixed weighted homoge-neous polynomial which satisfies H ( ρe iθ ◦ z ) = ρ d r e d p θ H ( z ) and thus 0 isthe only critical value. Thus ¯ ∂ (cid:60) H P ( w ) , ¯ ∂ (cid:61) H P ( w ) are linearly independentover R and we can proceed the same argument as above replacing ¯ ∂f, ¯ ∂g to ¯ ∂ (cid:60) H ( z ( t )) , ¯ ∂ (cid:61) H ( z ( t )) to conclude the real two dimensional subspace (cid:104) ¯ ∂ (cid:60) H ( z ( t )) , ¯ ∂ (cid:61) H ( z ( t )) (cid:105) R has a limit which is a real 2-dimensional subspaceof C J . Thus lim t → T z ( t ) V ( H, z ( t )) ⊃ C I . See the proof of Theorem 20, [21]for further detail. (cid:3) Now the proof of Theorem 7 is completed. (cid:3)
By Proposition 11, [21], we get the transversality assertion:
Corollary 13.
Let { f, g } be as in Theorem 7. Then there exists a positivenumber r such that for any r , < r ≤ r , there exists a positive number N THE MILNOR FIBRATION FOR f ( z )¯ g ( z ) II 15 δ ( r ) so that for any η (cid:54) = 0 with | η | ≤ δ ( r ) and and any ρ , r ≤ ρ ≤ r , thenearby fiber H − ( η ) is non-singular in B nr and intersects transversely withthe sphere S n − ρ . Existence of a tubular Milnor fibration.
By Lemma 2, Theorem7and Corollary 13, we apply Ehresmann’s fibration theorem ([35]) to obtain:
Theorem 14.
Assume that { f, g } satisfies the same assumption as in The-orem 7. Then there exists a positive number ε and a sufficiently small δ (cid:28) ε such that H = f ¯ g : E ( ε ; δ ) ∗ → D ∗ δ is a locally trivial fibration where E ( ε, δ ) ∗ := { ( z ) | (cid:54) = | H ( z ) | ≤ δ, (cid:107) z (cid:107) ≤ ε } and D ∗ δ := { ζ ∈ C | (cid:54) = | ζ | ≤ δ } , By Corollary 13, the fibration does not depend on the choice of ε and δ .For a disjoint locally tame non-degenerate pair { f, g } , applying the argumentof 2 of Example 5, we have: Corollary 15.
Asssume that { f, g } is a disjoint locally tame non-degeneratecomplete intersection pair. Fix an admissible toric modification ˆ π : X → C n .Take m > large so that { f m , g } and { f, g m } satisfy the toric multiplicitycondition with ˆ π . Then changing the coefficients of f m or g m slightly ifnecessary, { f m , g } and { f, g m } are locally tame non-degenerate completeintersections respectively for which ˆ π : X → C n satisfies the toric multiplicitycondition. Thus H m := f m ¯ g and K := f ¯ g m have tubular Milnor fibrations. For the definition f m , see Example 5.4. Spherical Milnor fibration
In this section, we study the existence of the spherical Milnor fibration.For a fixed small r >
0, we consider the mapping ϕ : S n − r \ K → S where K = V ( H ) ∩ S n − r and ϕ ( z ) = H ( z ) / | H ( z ) | . Lemma 16 (Lemma 10,[23]) . We assume { f, g } is a weak locally tame non-degenerate complete intersection pair satisfying the assumption in Theorem7. Then there exists a positive number r so that ϕ : S n − r \ K → S hasno critical points for any r, < r ≤ r . In the proof of Lemma 18 below, we will simultaneously reprove Lemma16. Using Lemma16 and the transversality property of the fibers H − ( η ) , (cid:54) = | η | ≤ δ and the sphere S n − r (Corollary 13), we obtain the following. Theorem 17.
Assume { f, g } is a weak locally tame non-degenerate completeintersection pair as in Theorem 7. For a sufficiently small r , ϕ : S n − r \ K → S is a locally trivial fibration.Proof. Consider the neighborhood of K defined by N ( K ) := { z ∈ S n − r \ K | | H ( z ) | ≤ δ } . Corollary 13 says that three vectors z , v ( z ) , v ( z ) are lin-early independent over R on N ( K ). For the definition of v , v , see the next section. Construct a vector filed V on S n − r \ K such that (cid:60) ( V ( z ) , v ( z )) = 1and furthermore if z ∈ N ( K ), it also satisfies (cid:60) ( V ( z ) , v ( z )) = 0. Thenalong the integral curves of V , the argument of H ( z ) is monotonic increaseand the absolute value of H is constant when it enters in the neighborhood N ( K ). Thus the integral curves exists for any time interval. For the localtriviality, we use the integration of V . (cid:3) Equivalence of tubular and spherical Milnor fibrations.
In thissection, we consider the equivalence problem of two Milnor fibrations. Letus recall two vector fields on the complement of V ( H ) which are defined asfollows ([19]). v = ∂ log H + ¯ ∂ log H = ∂f ¯ f + ∂g ¯ g v = i ( ∂ log H − ¯ ∂ log H ) = i (cid:18) ∂f ¯ f − ∂g ¯ g (cid:19) . v , v are real orthogonal. Let z ( t ) be a real analytic curve in C n \ V ( H ).Then we have ddt log H ( z ( t ))= 1 f ( z ( t )) n (cid:88) i =1 ∂f∂z i ( z ( t )) dz i ( t ) dt + 1¯ g ( z ( t )) n (cid:88) i =1 ∂g∂z i ( z ( t )) dz i ( t ) dt = 12 (cid:18) d z ( t ) dt , v ( z ( t )) − i v ( z ( t )) (cid:19) + 12 (cid:18) d z ( t ) dt , v ( z ( t )) + i v ( z ( t )) (cid:19) = (cid:60) (cid:18) d z ( t ) dt , v ( z ( t )) (cid:19) + i (cid:60) (cid:18) d z ( t ) dt , v ( z ( t )) (cid:19) . Thus we v ( z ) and v ( z ) are gradient vectors of (cid:60) log H ( z ) = log | H ( z ) | and (cid:61) log H ( z ) = i arg H ( z ). They are defined on C n \ V ( H ). A key lemma isthe following. Lemma 18.
Assume that { f, g } is as in Theorem 7. There exists a pos-itive number r such that for any z ∈ B nr \ V ( H ) , either three vectors z , v ( z ) , v ( z ) are linearly independent over R or they are linearly depen-dent and the relation takes the following form: z = λ v ( z ) + µ v ( z ) , λ, µ ∈ R , where λ is positive.Proof. Assume that there exists a real analytic curve z ( t ) in C n \ V ( H ) andreal valued rational functions λ ( t ) , µ ( t ) such that z ( t ) = λ ( t ) v ( z ( t )) + µ ( t ) v ( z ( t ))(7)and z (0) = . If µ ( t ) ≡
0, the assertion follows from Corollary 3.4, [15].Thus we may assume that µ ( t ) (cid:54)≡
0. Let I = { j | z j ( t ) (cid:54)≡ } . As f I , g I N THE MILNOR FIBRATION FOR f ( z )¯ g ( z ) II 17 satisfies the same assumption, we assume for simplicity that I = { , . . . , n } .Thus z ( t ) ∈ C ∗ n for t (cid:54) = 0. Consider their Taylor or Laurent expansions f ( z ( t )) = γt m f + (higher terms) , γ ∈ C ∗ g ( z ( t )) = βt m g + (higher terms) , β ∈ C ∗ z i ( t ) = a i t p i + (higher terms) , a i ∈ C ∗ λ ( t ) = λ t ν + (higher terms) , λ ∈ R µ ( t ) = µ t ν + (higher terms) , µ ∈ R ∗ . In the proof, we reprove Lemma 16. Thus λ = 0 only if λ ( t ) ≡ ν = + ∞ . If this is the case, z ( t ) is a critical pointof ϕ : S n − τ \ K τ where τ = (cid:107) z ( t ) (cid:107) and K τ is the link of H − (0) in thissphere. Put (cid:96) := min { d ( f ; P ) − m f , d ( P ; g ) − m g } and let us define ε f = (cid:40) , if d ( P ; f ) − m f = (cid:96) , if d ( P ; f ) − m f > (cid:96)ε g = (cid:40) , if d ( P ; g ) − m g = (cid:96) , if d ( P ; g ) − m g > (cid:96) As ord f ( z ( t )) ≥ d ( P ; f ) , ord g ( z ( t )) ≥ d ( P ; g ), we have that (cid:96) ≤
0. Put v k ( z ( t )) = ( v k ( t ) , . . . , v nk ( t )) for k = 1 ,
2. Observe that v j ( t ) = (cid:32) f P,j ( a )¯ γ ε f + g P,i ( a )¯ β ε g (cid:33) t (cid:96) − p j + . . . (8) v j ( t ) = i (cid:32) f P,j ( a )¯ γ ε f − g P,j ( a )¯ β ε g (cid:33) t (cid:96) − p j + . . . (9)Put P ( a ) := ( p a , . . . , p n a n ) , p min = min { p j | j ∈ I } ,J = { j | p j = p min } , ν = min { ν , ν } and put δ i = 1 or 0 according to ν i = ν or ν i > ν respectively for i = 1 , a j t p j + . . . = λ (cid:32) f P,j ( a )¯ γ ε f + g P,i ( a )¯ β ε g (cid:33) t ν + (cid:96) − p j + . . . + µ i (cid:32) f P,j ( a )¯ γ ε f − g P,j ( a )¯ β ε g (cid:33) t ν + (cid:96) − p j + . . . = e j t ν + (cid:96) − p j + . . . where e j = (cid:40) λ δ (cid:32) f P,j ( a )¯ γ ε f + g P,i ( a )¯ β ε g (cid:33) + µ δ i (cid:32) f P,j ( a )¯ γ ε f − g P,j ( a )¯ β ε g (cid:33)(cid:41) . If ν + (cid:96) − p min >
0, we get a contradiction a j = 0 , j ∈ J . Thus ν + (cid:96) − p min ≤
0. Consider the vectors v (0)1 = ( w , . . . , w n ) , w j = f P,j ( a ) γ ε f + g P,j ( a ) β ε g v (0)2 = ( w , . . . , w n ) , w j = i (cid:32) f P,j ( a ) γ ε f − g P,j ( a ) β ε g (cid:33) Assume that ν + (cid:96) − p min <
0. By (10), we get λ δ w j + µ δ w j = 0 , j = 1 , . . . , n. (10)If (cid:96) < ε f f P ( a ) = 0 and ε g g P ( a ) = 0. The above equality gives a contradic-tion to the non-degeneracy condition either for V ( f ) if ε f = 1 , ε g = 0, or for V ( g ) if ε f = 0 , ε g = 1 or for the intersection variety V ( f, g ) if ε f = ε g = 1.Assume (cid:96) = 0. Then ν < p min , γ = f P ( a ) and β = g P ( a ). We considerthe equality (cid:88) j ∈ J dz j ( t ) dt z j ( t ) = (cid:88) j ∈ J dz j ( t ) dt (cid:16) λ ( t ) v j ( t ) + µ ( t ) v j ( t ) (cid:17) . (11)The left hand side has order 2 p min − (cid:88) j ∈ J z j ( t ) dt z j ( t ) = n (cid:88) j ∈ J p j | a j | t p min − + . . . . Using (10) and Euler equality, we see that the leading term of the right handis t ν − which has the coefficient λ δ ( d ( P ; f ) + d ( P ; g )) + iµ δ ( d ( P ; f ) − d ( P ; g )) (cid:54) = 0 . The coefficient is non-zero. ( If δ = 0, we use the Newton multiplicitycondition to see the imaginary part is non-zero.) Thus the order is strictlysmaller than 2 p min −
1, which is a contradiction. Thus the case ν + (cid:96) − p min < ν + (cid:96) − p min = 0 . (10) implies the following equality. λ δ w j + µ δ w j = (cid:40) a j j ∈ J , j / ∈ J. We consider the equality (11) again.The left side of (10) has order 2 p min − (cid:80) j ∈ J p min | a j | >
0. The right side has order 2 p min − and the coefficient is given through Euler N THE MILNOR FIBRATION FOR f ( z )¯ g ( z ) II 19 equality as λ δ (cid:18) d ( P ; f ) f P ( a ) γ ε f + d ( P ; g ) g P ( a ) β ε g (cid:19) + µ δ i (cid:18) d ( P ; f ) f P ( a ) γ ε f − d ( P ; g ) g P ( a ) β ε g (cid:19) If (cid:96) < f P ( a ) ε f = g P ( a ) ε g = 0 and the above coefficient is zero. Thus weget a contradiction. Thus the only possible case is (cid:96) = 0 and therefore f P ( a ) , g P ( a ) (cid:54) = 0 , ν = 2 p min . We observe also δ (cid:54) = 0, as otherwise the coefficient is purely imaginary.Thus we should have (cid:96) = 0 , ν ≤ ν , γ = f P ( a ) , β = g P ( a ) . The leading coefficients of (11) gives the equality: (cid:88) j ∈ J p min | a j | = λ ( d ( P ; f ) + d ( P ; g )) + iδ µ ( d ( P ; f ) − d ( P ; g )) . Thus taking the real part of this equality, we conclude that λ >
0. Thisalso proves λ ( t ) ≡ λ (cid:54) = 0. This gives another proof ofCorollary 13. (cid:3) Now we are ready to prove the equivalence theorem.
Theorem 19.
Assume that { f, g } is a locally tame non-degenerate completeintersection pair as in Theorem 7. Consider the tubular and spherical Milnorfibrations H : ∂E ( r, δ ) ∗ → S δ ϕ : S n − r \ K → S . These two fibrations are equivalent. Here we use the notations ∂E ( r, δ ) ∗ := { z ∈ B nr | | H ( z ) | = δ } , K = S n − r ∩ V ( H ) . Proof.
Let δ be sufficiently small and put ∂N ( K ) := { z ∈ S n − r | | H ( z ) | <δ } . By the transversality, N ( K ) is contractible to K and N ( K ) \ K isdiffeomorphic to ∂N ( K ) × (0 ,
1) and ϕ : S n − r \ N ( K ) → S is equivalent tothe spherical fibration ϕ : S n − r \ K → S . Note that vectors v , v are realorthogonal. Take a locally finite open covering U = { U α , α ∈ A } ∪ { V β , β ∈ B } of B nr ∩{ z | | H ( z ) | ≥ δ } as follows. Each U α , V β are open disk with center p α , p β . Secondly in each U α , { z , v ( z ) , v ( z ) } are linearly independent over R , while in V β , p β can be written as p β = λ v ( p β ) + µ v ( p β ) , λ > V β is small enough so that (cid:60) ( z , v ( z )) > z ∈ V β . (There might exist a point z ∈ V β where z , v ( z ) , v ( z ) are linearly independent.) Construct a vector field w α on U α so that (cid:60) ( w α ( z ) , v ( z )) = 0 , (cid:60) ( w α ( z ) , v ( z )) = 1 , (cid:60) ( w α ( z ) , z ) = 1 , ∀ z ∈ U α . On V β , we simply take w β ( z ) = v ( z ). As V β is small enough, (cid:60) ( w β ( z ) , z ) > (cid:60) ( w β ( z ) , v ( z )) = 0 for ∀ z ∈ V β . Then glue together these vectorsusing a partition of unity to get a vector field X ( z ) on B nr ∩{ z | | H ( z ) | ≥ δ } .Note that for any z ∈ B nr ∩ { z | | H ( z ) | ≥ δ } , (cid:60) ( X ( z ) , v ( z )) = 0 and (cid:60) ( X ( z ) , z ) > (cid:60) ( X ( z ) , v ( z )) > z ( t )starting a point p ∈ ∂E ( r, δ ) ∗ , arg H ( z ( t )) is constant and | H ( z ( t )) | , (cid:107) z ( t ) (cid:107) are strictly increasing. This curve arrives at z ( s ( p )) ∈ S n − r \ N ( K ) for afinite time s ( p ) >
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