DOI 10.1007/s13163-013-0118-2
Motivic Milnor fibers of a rational function
Michel Raibaut
Received: 22 March 2012 / Accepted: 1 June 2012 / Published online: 6 April 2013
© Universidad Complutense de Madrid 2013
Abstract In this paper, following the approach of Denef and Loeser, we define motivic Milnor fibers of a rational function corresponding to its topological Milnor fibers considered by Gusein-Zade, Luengo and Melle–Hernández.
Keywords Algebraic Geometry·Singularities·Meromorphic function·Milnor fiber·Monodromy·Bifurcation set·Vanishing cycles·Mixed Hodge modules· Motivic integration·Motivic Milnor fiber
Introduction
Let d be a positive integer and P and Q be two relatively prime polynomials in C[x1,…,xd]. We denote byIthe common zero set of P and Q and by f the rational function P/Q well defined onAdC\ItoP1C. Gusein-Zade et al. [16,18] (in the more general meromorphic setting) studied the germ of f at an indeterminacy point x. They showed in particular that for all value a inP1C, there exists0positive such that for any positivesmaller than0, the function f from B(x, )\ItoP1Cis a topological locally trivial fibration over a punctured neighborhood of a. Thus, they defined a Milnor fiber of f at x for the value a, denoted by Fx,a, endowed with a monodromy action induced by the fibration and denoted by Tx,a. They proved, as A’Campo did it in the holomorphic case, a formula for the monodromy zeta function of Tx,aand its
This project has been supported by the Instituto de Matemática Interdisciplinar, by the ERC Advanced Grant NMNAG and the project ANR-08-JCJC-0118-01.
M. Raibaut (
B
)Université Pierre and Marie Curie, Institut de Mathématiques de Jussieu, 4 Place Jussieu, Case 247, 75252 Paris Cedex 5, France
e-mail: raibaut@math.jussieu.fr
Lefschetz numbers, in terms of a log-resolution of the zero set of P and the zero set of Q. A value a is called typical, and atypical otherwise, if the fibration can be extended as a trivial fibration over a neighborhood of a. The set of atypical values is finite and called the bifurcation set of the germ of f at x.
Gusein-Zade et al. [18] studied also the global situation and showed, as in the holomorphic case, that f is a topological locally trivial fibration over the complex projective line away from a finite subset. This allowed them to construct a “global Milnor fiber” of f for the value a, denoted by Ga, which carries again a monodromy action Ta. They also defined a bifurcation set in this context.
We mention also works by Siersma and Tibar [38,39] and more recently by Bodin et al. [3,4]. Finally, note that this problem is closely related to the study of pencils s P+t Q =0, we refer as example to Lé-Weber [24], Parusi´nski [33] and Siersma and Tibar [38].
The paper is organized as follows. In Sect.1 we explain the motivic setting and some constructions useful in the following. In Sect.2, we explain the topological point of view mentioned before. In Sect.3, we construct sheaf-theoretic interpretations of these topological situations. These constructions will also be defined at the level of mixed Hodge modules of Saito and will give the limit mixed Hodge structure on the cohomology groups of the Milnor fibers.
In the same way as in Denef and Loeser [7,10] and more recently in Guibert et al.
[15], we introduce in Sect.4motives associated to the above topological objects. In particular, we construct a motivic Milnor fiber of the germ of f at x for the value a and a global motivic Milnor fiber of f for the value a, taking account of singularities at infin- ity. As in the polynomial case [36], we obtain a motivic bifurcation set in the local and global situations. These objects contain additive and multiplicative invariants of the Milnor fibers, and as in [7] realize the classes in appropriate Grothendieck rings of the above mixed Hodge modules and limit mixed Hodge structures. We also obtain the for- mula for the Lefschetz numbers of Tx,ain [16]. These results were announced in [35].
1 Motivic setting
Bellow we explain some definitions and properties that will be used through out the paper. We refer to [9,14,15,25,26] for further discussion.
1.1 Grothendieck rings of varieties 1.1.1 Varieties
Let k be a field of characteristic 0 and denote byGmits multiplicative group. We call k-variety, a separated reduced scheme of finite type over k. We denote by V ark the category of k-varieties and for any k-variety S, by V arS the category of S-varieties, where objects are morphisms X →S in V ark.
Let G be an algebraic group and let X be a variety endowed with an action of G.
This action is said to be good, if every G−orbit is contained in an affine open subset of X .
Let X be a variety over k and let p : A→ X be an affine bundle for the Zariski topology. The fibers of p are affine spaces and the transition morphisms between affine charts are affine maps. Let G be an algebraic linear group. A good action of G over A is called affine if and only if it is a lifting of a good action of G over X and its restriction to all the fibers is affine. In the following we will suppose all the actions are good.
1.1.2 Grothendieck rings of varieties
Let S be a k-variety. Let X be an S-variety endowed with a good action ofGm, denoted byσ. A morphismπ: X→Gmis said to be homogeneous of weight n, ifπ(σ(λ,x)) is equal toλnπ(x)for allλinGmand for all x in X . For all positive integer n we denote byV arSG×Gm,n
m the category of varieties X → S×Gm endowed with a good action of Gm, such that the fibers of the projection to S areGm-invariants and the projection to Gmis homogeneous with weight n.
Definition 1 For all positive integer n, the Grothendieck ring of varieties, denoted by K0(V arSG×Gm,n
m), is defined in [15, Section 2] as the free abelian group generated by the isomorphism classes of varieties X →S×Gmof the category V arSG×Gm,n
m, modulo the following relations:
1.
[X→ S×Gm, σ] = [X→S×Gm, σ] + [X\X→S×Gm, σ] with Xa closed subset of X invariant underGm,
2.
[X×Ank →S×Gm, σ] = [X×Ank →S×Gm, σ]
ifσ andσare two liftings of the sameGm-action over X to affine actions on the bundle,
3.
[X → S×Gm, σ] = [X→ S×Gm, σl]
whereσl(λ,x)is equal toσ (λl,x)for allλinGm, all x in X , all positive integer l and for any actionσ ofGm over a variety X .
For all positive integer n, the fibered product above S×Gm endowed with the diagonal action, naturally induces a product in the category V arSG×Gm,n
m and a ring structure on the group K0(V arSG×Gm,nm). The unit for the product, denoted by 1S×Gm, is the class of the identity morphism on S×Gm where Gm acts trivially on S and by translation on itself. In particular, the ring K0(V arSG×Gm,nm)has a K0(V ark)-module structure.
For a nice survey on Grothendieck rings of varieties we refer to the paper of Nicaise and Sebag [31].
1.1.3 Localization
For each positive integer n, in the ring K0(V arGS×Gm,nm)we denote byLthe class[A1k× S×Gm, πS×Gm, τn]whereπS×Gm is the projection to S×Gm andτn(λ, (a,x, μ)) is equal to(a,x, λnμ)for all(a,x, μ)inA1k×S×GmandλinGm. Then we denote byMGS×Gm,nm the localized ring K0(V arGS×Gm,n
m)[L−1]. 1.1.4 Direct image and inverse image
Let f :S →Sbe a morphism of varieties. The composition by f induces the direct image group morphism
f!:MGS×Gm,nm →MGSm×G,nm.
The fibered product over Sinduces the inverse image ring morphism f−1:MGSm×G,nm →MGS×Gm,nm.
We can also define these morphisms at the level of K0. 1.1.5 Rational series
Let A be one of the ringsZ[L,L−1],Z[L,L−1, (1/(1−L−i))i>0]andMGS×Gm m. We denote by A[[T]]sr the A-submodule of A[[T]]generated by 1 and finite products of terms pe,i(T)= LeTi/(1−LeTi)with e inZand i inN>0. There is a unique A-linear morphism limT→∞ : A[[T]]sr → A such that for any subset(ei,ji)i∈I of Z×N>0with I finite or empty, limT→∞(
i∈I pei,ji(T))is equal to(−1)|I|. 1.1.6 Polyhedral convex cone
Let I be a finite set. A rational polyhedral convex cone ofR∗|I| is a convex part of R∗|I|defined by a finite number of linear inequalities with integer coefficients of type a≤0 and b>0 and stable under multiplication by elements ofR>0.
1.1.7 Rationality lemma
The following lemma is well known [13, 2.1.5] and [15, 2.9].
Lemma 1 Let I be a finite set. Letbe a rational polyhedral convex cone inR|>I|0. We denote bythe closure ofinR|≥I|0. Let l andνbe two linear forms onZ|I|,with l andνpositive on\ {0}. In the ringZ[L,L−1][[t]], we denote by S,l,ν(t)the series
k∈∩N|I|>0tl(k)L−ν(k).
Whenis open in its linear span andis generated by a family of vectors that can be extended as aZ−basis of theZ-moduleZ|I|, the series S,l,ν(T)are rational and their limit limT→∞S,l,ν(T)is equal to(−1)di m().
In the general case, by additivity with respect to disjoint union of cones with the positivity assumption, one deduces that the series S,l,ν(T)are rational and their limit is equal to the Euler characteristic with compact support of the cone.
1.2 Resolutions
Let X be a smooth variety with pure dimension d, and let Z be a closed subset of X with codimension everywhere bigger than 1. A log-resolution h : Y → X of (X,Z)is a proper morphism h : Y → X with Y smooth such that the restriction h :Y\h−1(Z)→X\Z is an isomorphism and h−1(Z)is a normal crossing divisor.
We will always suppose that the irreducible components of the divisor are smooth.
Hironaka [19] proved the existence of a log-resolution.
1.2.1 Stratification of the exceptional divisor
We denote by (Ei)i∈A the irreducible components of the divisor h−1(Z). For any subset I of A, we denote by EI the intersection∩i∈IEi and by E0I the constructible set∩i∈IEi \ ∪j∈/IEj while E∅0is Y \h−1(Z).
1.2.2 Divisors
If I is a sheaf of ideals defining a closed subscheme Z and h−1(I)OY is locally principal, then we define the multiplicity of I along Ei, denoted by Ni(I)as the coefficients in the equality
h−1(Z)=
i∈A
Ni(I)Ei.
IfIis a sheaf of principal ideals generated by a function g, we will write Ni(g) instead of Ni(I). In the same way we defineνi by the equality
divJ ac h=
i∈A
(νi−1)Ei
where J ac h is the jacobian ideal of h.
1.2.3 UI
For all i in A, we denote byνEithe normal bundle of Eiin Y and by UEithe complement of the zero section in the bundleνEi. For any non empty subset I of A such that EIis non empty, we denote byνI the fiber product of the restriction to EI of the bundles νEi with i in I ,UI the fiber product of restrictions to E0I of the bundles UEi with i in I andπ the canonical projections of the bundles to their bases.
1.2.4 Structural application fI :UI →Gm
Let X be a smooth variety with pure dimension d and let f be a morphism on X . We suppose the special fiber f−1(0)is nowhere dense in X and we denote it by X0(f). Let F be a reduced divisor containing X0(f), and h:Y → X be a log-resolution of (X,F). We denote by E the exceptional divisor, by A the index set of the irreducible components Ei of E and by Ni(f)the multiplicities of the divisor div(f ◦h). We fix a non-empty subset I of A such that for some i in I , Ni(f) > 0, in particular h(E0I)is contained in X0(f). Following [15, 3.4, 3.5] and [14, 2.6], we describe here how UI is endowed with a morphism fI toGm and an action σ of Gm such that (UI, (h◦πI, fI), σI)belongs to V arGXm
0(f)×Gm.
Since the multiplicative groupGm naturally acts on each UEi, the diagonal action induces aGm-actionσon UI. The morphism fIcan be defined thanks to a deformation to the normal cone of EIin Y [12]. We consider the affine spaceA|kI|as Spec k[(ui)i∈I] and the subsheafOY×A|I|
k [(u−i 1)i∈I]. Let AI :=
n∈N|I|
OY×A|I|
k
−
i∈I
ni(Ei ×A|kI|)
i∈I
u−i ni.
We denote by CYI the spectrum SpecAI.The natural inclusionOY×A|I| k inAI
induces a morphism
πI :CYI →Y ×A|kI|
and then, by composition with the projection, a morphism pI :CYI →A|kI|.The ring AIis a graded subring ofOY[(ui,u−i 1)i∈I]. We consider the actionσIofGmI on CYI
under which the sections ofOY are invariant and acting by(λi,ui)→λi−1ui. Then, we have the following lemma [15, 3.5, Lemme 5.12 ]:
Lemma 2 The bundleνEI is smooth and is identified, in a equivariant way, with the fibre p−I1(0). The image of the function f◦h by the inclusion ofOY×A|I|
k inAIcan be divided by
i∈IuiNi(f). We denote by f˜I the quotient inAI. The restriction of f˜I to the fiber p−I1(0)is the function fI. This function does not vanish on UI and induces a smooth monomial morphism fI :UI → {0} ×Gm.
1.3 Arcs 1.3.1 Arc spaces
Let X be a k-variety. We denote byLn(X)the space of n-jets of X . This set is a k- scheme of finite type and its K -rational points are morphisms Spec K[t]/tn+1→ X ,
for any extension K of k. There are canonical morphismsLn+1(X)→Ln(X). These morphisms areAdk-bundles when X is smooth with pure dimension d. The arc space of X , denoted byL(X), is the projective limit of this system. This set is a k-scheme and we denote byπn : L(X)→ Ln(X)the canonical morphisms. For more details we refer as an example to [8,26].
1.3.2 Origin, order, angular component and action
For an elementϕin K[[t]]or in K[t]/tn+1, we denote by ord(ϕ)the valuation ofϕand by ac(ϕ)its first non-zero coefficient. By convention ac(0)is zero. The scalar ac(ϕ) is called the angular component ofϕ. The multiplicative groupGm acts canonically onLn(X)and onL(X)by
λ.ϕ(t):=ϕ(λt).
We consider the application originϕ→ϕ(0)=ϕ mod t 1.3.3 Contact order
Let X be a variety and let F be a closed subscheme of X . We denote byIF the ideal of functions which vanish on F . We denote by ord F the function which assigns to each arcϕinL(X)the bound inf ord g(ϕ)where g belongs to all the local sections of IFat the origin ofϕ.
1.3.4 A rewrite lemma
In the following we will use the lemma [8, Lemma 4.2] which follows from Hensel lemma.
Lemma 3 Let X be a smooth variety with pure dimension d, let U be an affine open subset of X , let:U →Adbe an étale map. In the category V ark, for all integer n, induces the isomorphism
n:Ln(U)→U×Ad Ln(Ad)
which associates to an n-jetϕ, the couple(ϕ(0), ◦ϕ), where U×AdLn(Ad)is the fiber product ofand the application “origin of arcs”. Taking inductive limit, this induces an isomorphism :L(U)→U×Ad L(Ad)that associates to an arcϕthe couple(ϕ(0), ◦ϕ).
1.4 The motivic Milnor fiber morphism
Let X be a variety and let f : X →A1k be a morphism.Using the weak factorisation theorem, Bittner [2] extends the motivic Milnor fiber as a morphism defined over all the Grothendieck ringM . Guibert, Loeser and Merle [15] give a different construction
using the motivic integration theory introduced by Kontsevich in [21]. Below, we explain this construction.
Definition 2 Let X be a smooth variety with pure dimension d, let U be an open and dense subset of X , let F be its complement and let f :X →A1kbe a morphism. Let n andγ be two positive integers, we consider the arc space
Xγn(f):=
ϕ ∈L(X)|ord f(ϕ)=n,ordϕ∗F≤nγ
endowed with the arrow “origin, angular component”(π0,ac(f))to X0(f)×Gm
and the standard action ofGm on arcs. Then, we consider the modified motivic zeta function
Zγf,U(T):=
n≥1
μ(Xγn(f))Tn∈MGXm0(f)×Gm[[T]].
As in Denef and Loeser, there is a rationality result [15, Section 3.8]
Proposition 1 Let U be an open and dense subset of a smooth variety X with pure dimension d. Let f : X→A1kbe a morphism. There is an integerγ0such that for all integerγbigger thenγ0, the series Zγf,U(T)are rational and their limit is independent ofγ. We will denote bySf,U the limit−limT→∞Zγf,U(T). Furthermore, if X0(f)is nowhere dense in X , and(Y,h)is a log-resolution of(X,F∪X0(f))then there is a direct image formula
Sf,U = −
I=∅
I⊂C
(−1)|I|[UI] =h!(Sf◦h,h−1(U))∈MGXm0(f)×Gm,
where with notations of Sect.1.2, C is the set of{i ∈ A|Ni(f)=0}.
Then, Bittner and in a different way Guibert, Loeser and Merle proved the extension theorem to all the Grothendieck group ([2], [15, Section 3.9]):
Theorem 1 Let X be a variety and f : X → A1k a morphism. There is a unique Mk-linear group morphismSf : MX → MGXm0(f)×Gm such that for all proper morphism p : Z → X with Z smooth, and for all open and dense subset U in Z, Sf([p:U → X])is equal to p!(Sf◦p,U).
Remark 1 Let X be a variety, U ⊂X be an open and dense subset, F its complement and f : X → A1k a morphism. If f is smooth over X then Sf(X)is equal to the class[X0(f)×Gm,i d, τ],whereτ is the translation. If F is a smooth closed subset of X , the injection i : F → X is proper so the image Sf([i : F → X])is equal to i!Sf◦i(F)which is i!Sf|F. By additivity, the result is also true for any closed subset of X . Consequently, if f is smooth over X and its restriction to F is also smooth then Sf(U)is equal to[(U∩X0(f))×Gm,i d, τ],whereτ is the translation.
Definition 3 Let X be a variety and let f :X →A1kbe a morphism. Let U be an open subset of X and let F be the complement. The motivic nearby cycles of f relative to U are the objectsSf([U])and the motivic vanishing cycles are the objects
Sf([U]):=(−1)dim X−1(Sf(U)− [(U∩X0(f))×Gm,i d, τ]).
These motives are elements of the ringMGX0m(f)×Gm. By the above remark, if f is smooth over X and the restriction to F of f is smooth then the motivic vanishing cyclesSf([U])are zero.
1.5 Realizations
The discussion in this subsection is based on [9], [15, 3.16, 6.1] and [34]. We suppose here k=C.
1.5.1 Hodge realization
We will denote by H S the abelian category of Hodge structures and by K0(H S)its associated Grothendieck ring. Recall that a mixed Hodge structure V with weight fil- tration W•has a canonical class[V] :=
m[GrmW(V)]in the ring K0(H S),where for all m, [GrmW(V)] is the class of the Hodge structure of the mth-graded part Wm(V)/Wm−1(V). There is a canonical morphism
χh :MC→ K0(H S)
which associates to the class [X] of a variety X the element
i(−1)i[Hci(X,Q)]in K0(H S), where for all i the class[Hci(X,Q)]is the class of the mixed Hodge structure of Deligne on Hci(X,Q).
We will denote by H Smonthe abelian category of Hodge structures endowed with a quasi-unipotent endomorphism and K0(H Smon)its Grothendieck ring. There is a canonical morphism
χh :MGGmm →K0(H Smon)
defined as follows. If[X]is the class of f : X → Gm inMGGmm with X connected, and f is homogeneous toward the actionσ ofGm, then f is a locally trivial fibration in the complex topology. Furthermore, if the weight is equal to n then for all x in X the function
[0,1] → X
t →σ(ex p(2iπt/n),x)
is a geometrical monodromy with finite order around the origin. The fiber of f at 1, denoted by X1is endowed with an automorphism with finite order Tf, and we define
χh([f :X →Gm, σ]):=
i
(−1)i[Hci(X1,Q),Tf]
.
There is a linear function called Hodge spectrum
sph: K0(H Smon)→Z[Q] := ∪n≥1Z[t1/n,t−1/n] [H] sph([H]):=
α∈Q∩[0,1[tα(
p,q∈Zdim Hα(p,q)tp) where for a Hodge structure with quasi-unipotent operator, Hαp,qis the characteristic space of Hp,qof the eigenvalue e2iπα. We denote by Sp the composition sph◦χh. 1.5.2 Mixed Hodge modules realization
For the definitions and results described below we refer to [34, Chapter 14]. For any complex algebraic variety X , we denote by M H MX the abelian category of mixed Hodge modules over X and we denote by K0(M H MX)the associated Grothendieck ring. We consider M H MmonX the abelian category of mixed Hodge modules on X endowed with an automorphism with finite order and K0(M H MmonX )its associated Grothendieck ring.
Connection with perverse sheaves. By definition of mixed Hodge modules, there is a faithful functor r atX : Db(M H M(X)) → Dcb(X,Q) such that the category M H M(X)corresponds to the category Perv(X,Q). For any mixed Hodge module M on X , we call r atXM its underlying rational perverse sheaf. For this part we refer to [34, 14.1:Axiom A].
Connection with mixed Hodge structures By construction of mixed Hodge modules, there is an equivalence of categories between the category M H MSpecmonC of mixed Hodge modules supported by the point and endowed with a quasi-unipotent endo- morphism and the category M H Smon of mixed Hodge structures endowed with a quasi-unipotent endomorphism. Then, we will denote by: K0(M H M)monSpecC → K0(H S)monSpecCthe ring morphism induced by the canonical application which asso- ciates to a mixed Hodge structure endowed with a quasi-unipotent morphism its class in the ring K0(H Smon). For this part we refer to [34, 14.1:14.61: Axiom B].
Mixed Hodge modules realization There is a realization morphism between varieties and mixed Hodge modules:
χM H M :K0(V arX)→ K0(M H MX).
Indeed, by additivity, there is a unique morphism of ringsχM H M such that for all p : Z → X with Z smooth, χM H M([p : Z → X])is the class of the direct image with compact support[p!QZ(−1)]. Furthermore, this morphism extends to a Mk-linear morphism
χM H M :MX →K0(M H MX)
with K0(M H MX)considered as aMSpecC-module with its structure of module over K0(M H MSpecC), and the Hodge realization
χM H M :MSpecC→K0(M H MSpecC).
There is also a natural morphism compatible in the equivariant case ([15, 2.6, 3.16]) χM H M :MGX×Gm m → K0(M H MXmon).
Note that the composition betweenandχM H M is the ring morphismχh.
Nearby cycles Let f : X →A1Cbe a function. Saito [37] constructs a nearby cycles functor f and a vanishing cycles functor f from the category M H MX to the category M H MmonX
0(f) such that the underlying perverse sheaves r atXf(QX) and r atXf(QX)are equals toψf(QX)[−1]andφf(QX)[−1]. Then, we have the nearby cycle functor
f :K0(M H MX)→K0(M H MmonX
0(f)).
Compatibility between the motivic Milnor fiber morphism and the nearby cycles func- tor Denef and Loeser [7, 4.2.1], and Guibert et al. [15, 3.17] showed the following compatibility theorem between the motivic Milnor fiber morphism and the nearby cycle morphism, modulo the Hodge realization:
Theorem 2 The following diagram is commutative:
MX
Sf //
H
MGXm0(f)×Gm
H
K0(M H MX) f //K0(M H MmonX0(f))
By composition with the morphism r atX, we obtain a morphism MX → K0(PervmonX0 )
such that Sf(X)and Sφf(X)realize the classes[ψf(QX)[−1]]and[φf(QX)[−1]]
and for any open and smooth subset U of X , Sf(U)and Sφf(U)realize as the classes [ψf(Ri!QU)[−1]]and[φf(Ri!QU)[−1]]where i :U→ X is the open immersion.
2 Topological point of view
2.1 Setting and notations
Let d be a positive integer, let P and Q be two relatively prime polynomials in C[x1,…,xd]. Let f be the rational function P/Q and denote by I(f)the indeter- minacy locus
I(f):= {x∈Cd| P(x)=Q(x)=0}.
Thus, we have a well defined function
f :Cd\I(f)→P1C x→ P(x)
Q(x)
In this section, we describe the topological point of view in the local case and in the global case.We follow here the articles of Gusein-Zade et al. [16,18].
2.2 Local point of view
Theorem 3 [16,18] Let x be an indeterminacy point.For all value a inP1C, there is ε0 >0 such that for allε≤ ε0, the sphere S2d−1(a, ε)is transverse to all strata of the fiber f−1(a)and the function f : B2d(x, ε)\ I(f) → P1Cis a locally trivial topological fibration over a punctured neighborhood of a.
Definition 4 The above fibration is called Milnor fibration of the germ f at x for the value a. One of its fiber
Fx,a:= {z∈ B2d(x, ε)\I(f), f(z)=P(z)/Q(z)=a}
forεsmall enough and asufficiently close to a, is a smooth non compact complex variety with boundary of dimension d−1 and is called the Milnor fiber of the germ
f at x of the value a.
Definition 5 The Milnor fibration of the germ f at the point x for the value a induces a monodromy transformation
Tx,a: Fx,a→ Fx,a
which is well defined up to isotopy and called the monodromy transformation.
Definition 6 A value a inP1Cis said to be typical for the germ f at x, if and only if for allεsmall enough the function f : B2d(x, ε)\I(f)→P1Cis a trivial fibration over a neighborhood of a, otherwise a is said to be atypical. The bifurcation set B(f,x)of
f at the germ x is the set of atypical values.
Remark 2 If a value c is typical then the monodromy transformation is isotopic to the identity. Furthermore:
Theorem 4 [18] There is a finite setofP1Csuch that for all c inP1C\the Milnor fibers of f at x are diffeomorphic and the monodromy transformations are isotopic to the identity. In particular the number of atypical values is finite.
2.3 Global point of view
As in the polynomial case, following ideas of Thom, we have the theorem
Theorem 5 [18] The application f :Cd\I(f)→P1Cis a locally trivial topological fibration over the complement of a finite set of the projective line.
Definition 7 Any fiber of this fibration is called a generic fiber of f . The smallest set B(f)such that f is a topological locally trivial fibration over B(f)\P1Cis called the bifurcation set of f . Its elements are called atypical values.
Remark 3 The set of atypical values contains the critical values of f but also other values that come from singularities at infinity. In the meromorphic case, we refer for instance to the article of Gusein-Zade et al. [17]. In the holomorphic case, we refer to articles of Broughton [5], Cassou-Noguès and Dimca [6], Kurdyka et al. [22], Némethi and Zaharia [28,29], Parusi´nski [32], Tib˘ar [40,41] and Zaharia [42].
In order to take account of singularities at infinity, we will take below a compacti- fication(Xˆ,i, fˆ)of(Cd\I, f).
Definition 8 By the above fibration theorem, for a value a, f is a locally trivial topological fibration over a punctured neighborhood of a small enough. The fiber of this fibration is called Milnor fiber of f for a value a, denoted Ga, as a “global” Milnor fiber. The associated monodromy Tais the monodromy of f at the value a.
Remark 4 The Milnor fibers Fx,aand Gaare well defined from the topological point of view, but not from the algebraic point of view. In the next section we will consider the sheaf analog, which will be well defined in the algebraic setting.
3 Sheaf point of view
Consider the following commutative diagram:
Cd\I //
fUUUUUUUUUUUUU**
UU UU UU UU UU
U X⊂(Cd\I)×P1C i //
p
X⊂Cd×P1C f
wwnnnnnnnnnnnnnn
ˆ
i //Xˆ
ˆ
ssggggggggggggggggggf ggggggggggggggg
P1
where X is the graph of f in(Cd\I)×P1C, X is its Zariski closure inCd×P1C, i is the associated closed immersion, p and f are the projections onP1C,X is a variety,ˆ iˆ is an open and dominant immersion and f is a proper map. We will denote by i theˆ composition of immersionsˆi◦i , by F the complementary of X in X , and by F theˆ complementary of X inX .ˆ
Remark 5 The varieties X andX are possibly not smooth and hence the closed subsetsˆ F andF may contain singularities.ˆ
3.1 Local situation
Let x be an indeterminacy point of f and let a be a value.
Remark 6 In this singular situation, Lê Dung Trang in [23] defined a Milnor fiber of f at (x,a)which is topologically identical to the Milnor fiber Fx,a considered by Gusein-Zade et al. [16,18].
Definition 9 Let Fa be the complex of constructible sheaves in the category Dbc(f−1(a))mon defined byψf−a(Ri!QX)(where f −a is 1/f if a is infinite). The complex of sheaves associated to the Milnor fiber Fx,a is the germFx,a ofFaat (x,a)naturally endowed with a monodromy action induced by the construction of the nearby cycles sheaf [11, Section 4.2].
This sheaf construction is connected to the topological point of view [11, Proposi- tion 4.2.2]:
Proposition 2 For all k, there exists an isomorphism between the cohomology groups
Hk(Fx,a)Hck(Fx,a,Q)Tx,a endowed with their monodromy action.
3.2 Global situation
Let a be a value. We consider now the complex of constructible sheaves associated to the global Milnor fiber Ga. In order to take account of singularities at infinity, we use the compactification(Xˆ,i, fˆ)of(X,p)or(Cd\I,f).
Definition 10 Let a be a value. We consider the direct image over the point of ψfˆ−a(Ri!QX):
Ga:=Rfˆ!ψfˆ−a(Ri!QX)∈Dbc({a})mon.
As in the polynomial case this sheaf construction is associated to the “global”
Milnor fiber Gaand will not depend on the compactification.
Proposition 3 The complex of constructible sheavesGadoes not depend on the com- pactification(Xˆ,iˆ, fˆ)and for all k, there exists an isomorphism betweenHkGaand Hck(Ga,Q)Ta for all k.
Proof We only prove the independence of compactification, since the isomorphism between the cohomology groups of the complex of sheaf and the cohomology groups is classical [11]. Let(Xˆ1,ˆi1, fˆ1)and(Xˆ2,ˆi2, fˆ2)be two compactifications and denote by Fˆ1 and Fˆ2 the complement of X . From a well known application of Hironaka theorem (see [36, Theorem 2.8]) there is(Z,E,h1,h2)such that,(Z,E,h1)is a log- resolution of(Xˆ1,Xˆ1\X)and(Z,E,h2)is a log-resolution of(Xˆ2,Xˆ2\X)such that we have the following commutative diagram
(Z,E)
h1
{{xxxxxxxx h2
##F
FF FF FF F
X1
ˆ f1
66
6666 6666 6666 X
f
i1
oo i2 // X2
ˆ f2
P1C
that is, we have the equalities fˆ1◦h1= ˆf2◦h2and j:=h−11◦i1=h−21◦i2. We obtain the equality from the direct image formula of Deligne:
Rfˆ1!ψfˆ1−a(Ri1!QX) =Rfˆ1!Rh1!ψ(fˆ1−a)◦h1(R j!QX)
=R(fˆ1◦h1)!ψ(fˆ1−a)◦h1(R j!QX)
=R(fˆ2◦h2)!ψ(fˆ2−a)◦h2(R j!QX)=Rfˆ2!ψfˆ2−a(Ri2!QX).
3.3 Mixed Hodge modules setting and limit mixed Hodge structures
These constructions exist at the mixed Hodge modules level of Saito [37]. Thus, we obtain the complex of mixed Hodge modulesFxMHM,a over{(x,a)}and the complex of mixed Hodge modulesGaMHMover{a}. By the same arguments they do not depend on the compactification.
The underlying mixed Hodge structures of HkFxMHM,a and HkGaMHM are by Propositions 2 and 3 the limit mixed Hodge structures of Hck(Fx,a,Q)Tx,a and Hck(Ga,Q)Ta. We refer to Sect.1.5.2 for a summary and [34] for a good intro- duction.
4 Motivic point of view
In this section we work with a field k of characteristic zero, and in the topological case, k isC. So we can suppose P and Q as polynomials of k[x1, ..,xd]with no constant common factor.
We will construct motives assigned to the complex of constructible sheaves Fx,a=ψf−a(Ri!QX)andGa=Rfˆ!ψfˆ−a(Ri!QX).
We will use constructions of Guibert–Loeser–Merle [15] mentioned in Sect.1.First, in Sect. 4.1we will construct a global motive Sfˆ,X above Xˆ ×Gm. Its restriction Sf,x,ato(x,a)×Gmin Theorem6, will be the motivic Milnor fiber of f for the value a at the point x whereas its direct image toP1×Gm followed by its restriction to {a} ×Gmwill be the motivic Milnor fiber Sf,aof f for the value a in Sect.4.3. These motives do not depend on the compactification. The Hodge realization theorem2of Denef–Loeser will imply that these objects are motivic analog of the above sheaves (Theorems8and11). Finally, in the local case we recover the Lefschetz numbers of the monodromy (Theorem9).
4.1 Global motivic zeta function
Below, we use notations of Sect.3. As in [36, Section 4] we consider the following definitions:
Definition 11 For two positive integers n andγ, we define
Xˆnγ := {ϕ∈L(Xˆ)|ordϕ∗Fˆ ≤nγ,ord[ ˆf(ϕ(t))− ˆf(ϕ(0))] =n}.
This semi-algebraic part of the arc spaceL(Xˆ)ofX is endowed with the usual actionˆ ofGmon arcs Sect.1.3.2, and the morphism “origin, angular component”
ϕ → [ϕ(0),ac(fˆ(ϕ)(t)− ˆf(ϕ)(0))] ∈ ˆX×Gm, where fˆ−a is 1/f when a isˆ ∞.
Definition 12 We consider the global motivic zeta function Zγˆ
f,X(T):=
n≥1
μ(Xˆγn)Tn∈MGˆm
X×Gm[[T]].
Remark 7 Let a be a value and consider Zγˆ
f,a,X(T)the restriction of Zγˆ
f,X(T)to fˆ−1(a)×Gm, namely we only use arcsϕwith fˆ(ϕ(0)) =a. By Proposition1, for γ big enough this series is rational and has a limit independ ofγ denoted Sfˆ,a,X.We define, as in Definition3, the motivic vanishing cycles of f relatively to f for theˆ value a by
