Singular cohomology of the analytic Milnor fiber, and mixed Hodge structure on the nearby cohomology
aa r X i v : . [ m a t h . AG ] S e p SINGULAR COHOMOLOGY OF THE ANALYTIC MILNORFIBER, AND MIXED HODGE STRUCTURE ON THE NEARBYCOHOMOLOGY
JOHANNES NICAISE
Abstract.
We describe the homotopy type of the analytic Milnor fiber interms of a strictly semi-stable model, and we show that its singular cohomologycoincides with the weight zero part of the mixed Hodge structure on the nearbycohomology. We give a similar expression for Denef and Loeser’s motivicMilnor fiber in terms of a strictly semi-stable model.MSC 2000: 32S30, 32S55, 14D07, 14G22 Introduction
Let k be any field, and let X be a k -variety, endowed with a morphism f : X → Spec k [ t ]. Let x be a closed point on the special fiber X s = f − (0). The analyticMilnor fiber F x of f at x was introduced and studied by Julien Sebag and theauthor in [20, 21, 22]. The object F x is defined as the generic fiber of the specialformal scheme b f : Spf b O X,x → Spf k [[ t ]]It is an analytic space over the field of Laurent series k (( t )), and serves as a non-archimedean model of the classical topological Milnor fibration (for k = C and X smooth). The arithmetic and geometric properties of F x reflect the nature of thesingularity of f at x . For instance, the ℓ -adic cohomology of F x is canonicallyisomorphic to the ℓ -adic nearby cohomology of f at x [6, 3.5], and the local motiviczeta function of f at x can be realized as a “Weil generating series” of F x [19, 9.7].If X is normal at x , then F x is a complete invariant of the formal germ ( f, x ) [19,8.8].In the present article, we study the topology of F x , considered as a k (( t ))-analytic space in the sense of Berkovich [4]. We denote by \ k (( t )) a the completionof an algebraic closure of k (( t )). Our first main result (Theorem 4.10) describes thehomotopy type of F x := F x b × k (( t )) \ k (( t )) a in terms of a so-called strictly semi-stable reduction of the germ ( f, x ). The secondmain result (Theorem 5.7) states that, if k = C , the singular cohomology of F x coincides with the weight zero part of the mixed Hodge structure on the nearbycohomology of f at x [25][17].These are local analogs of results by Berkovich [3]. He showed that the weightzero part of the limit mixed Hodge structure of a proper family Y → Spec C [ t ]coincides with the singular cohomology of the nearby fiber b Y η × C (( t )) \C (( t )) a , where b Y denotes the t -adic completion of Y and b Y η its generic fiber. Our proofs closely follow the ideas in [3], but we need additional work to pass from the global situationto our local one. The most important tool in our arguments is Berkovich’ descriptionof the homotopy type of the generic fiber of a poly-stable formal scheme [7].The above theorems have natural motivic counterparts. We give an expressionfor Denef and Loeser’s motivic Milnor fiber of f at x in terms of a strictly semi-stable model (Theorem 6.11). As we showed in [19], this motivic Milnor fiber canbe realized as a “motivic volume” of the analytic Milnor fiber F x . The expressionin Theorem 6.11 is similar in spirit to our description of the homotopy type of F x but uses different techniques (motivic integration) and yields another type ofinformation (class in the Grothendieck ring).Let us give a survey of the structure of the paper. In Section 2, we recall somebasic notions about special formal schemes and their generic fibers, and about theanalytic Milnor fiber. Section 3 contains the main technical tools of the paper: wedefine the simplicial set associated to a strictly semi-stable k -variety X , and weexplain how it can be used to describe the homotopy type of the generic fiber of astrictly semi-stable formal scheme X , following results by Berkovich. Moreover, weprove a crucial result about the homotopy type of the generic fiber of the completionof X along a union of irreducible components (Proposition 3.7 and its Corollary3.8). It is this property that allows to pass from Berkovich’ global situation to ourlocal one.In Section 4, these tools are applied to establish homotopy-equivalences betweencertain analytic spaces, and to study the homotopy type of the analytic Milnor fiber(Theorem 4.10). The key result is Proposition 4.4, which we now briefly explain.Any special formal k [[ t ]]-scheme X can be considered as a special formal k -scheme X k , by forgetting the k [[ t ]]-structure. The generic fiber of X k (where k carriesthe trivial absolute value) is naturally fibered over [0 ,
1[ by evaluating the points ofthe generic fiber (which are multiplicative semi-norms) in t . Proposition 4.4 studiesthe homotopy type of the fibers of this family.Section 5 contains the results concerning the mixed Hodge structure on thenearby cohomology of f at x (where k = C ). We recall Peters and Steenbrink’sconstruction of this mixed Hodge structure, and we show how the weight zero partcan be computed on a strictly semi-stable reduction (Proposition 5.6). Combiningthis result with the ones in Section 4, we see that the singular cohomology ofthe analytic Milnor fiber coincides with the weight zero part of the mixed Hodgestructure on the nearby cohomology of f at x (Theorem 5.7).In the final Section 6, we compare the preceding results with the motivic setting,and give an expression for Denef and Loeser’s motivic nearby cycles and motivicMilnor fiber in terms of a strictly semi-stable model (Theorem 6.11 and Corollary6.12). 2. Preliminaries
Some notation. If S is any scheme, a S -variety is a separated reduced schemeof finite type over S .For any locally ringed space Y , we denote by | Y | its underlying topologicalspace; if no risk of confusion arises, we’ll omit the vertical bars and we’ll denotethe underlying topological space by Y , to simplify notation. INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 3
For any field F , we denote by F a an algebraic closure, and by F s the separableclosure of F in F a . If L is a non-archimedean field (we do not exclude the trivialvaluation), then the absolute value on L extends uniquely to an absolute value on L a , and we denote by c L s and c L a the completions of L s , resp. L a (these completionscoincide if the valuation is non-trivial). We will work in the category of L -analyticspaces as introduced by Berkovich [5]. For any L -analytic space X , we put X = X b × L c L a .If R is a discrete valuation ring, with residue field k , and Y is a scheme over R ,then we denote its special fiber Y × R k by Y s .If T is any topological space and S is a subspace of T , then a strong deformationretract of T onto S is a continuous map φ : T × [0 , → T such that φ ( · ,
0) is theidentity, φ ( s, r ) = s for any point s of S and any r ∈ [0 , φ ( · ,
1) is a surjectiononto S .2.2. Strictly semi-stable (formal) schemes.
Let K be a complete discretelyvalued field, with valuation v K : K ∗ → Z . We do not assume that the valuation isnon-trivial, but we will assume that it is normalized (i.e. v K ( K ∗ ) = Z or v K ( K ∗ ) = { } ). We denote by K o the ring of integers in K , and by e K the residue field. If v K is trivial, then K = K o = e K . For each r ∈ ]0 , | · | r on K by | x | r = r v K ( x ) and we denote by K r = ( K, | · | r ) the correspondingnon-archimedean field. Moreover, we put K = ( e K, | · | ) where | · | is the trivialabsolute value. The distinction between K (which only carries a discrete valuation)and the fields K r (which are non-archimedean fields) is crucial for the applicationsin this article.If L = ( L, | · | L ) is any non-archimedean field, then we denote by L o the ringof integers and by e L the residue field. In particular, K or = K o and f K r = e K for r ∈ ]0 , X over L o is called stf t if it is separated, andtopologically of finite type over L o . We denote by X η its generic fiber, a L -analyticspace, and by X s its special fiber, a separated scheme of finite type over e L . Thereis a natural specialization map (of sets ) sp X : X η → X s . If X is a separatedscheme of finite type over L , we denote by X an the L -analytic space associated to X via non-archimedean GAGA [4, 3.4-5].A special formal K o -scheme is a separated adic Noetherian formal scheme X ,endowed with a morphism X → Spf K o , such that X / J is of finite type over K o for any ideal of definition J on X (this definition is slightly more restrictivethen the one used by Berkovich in [6]). In particular, any stf t formal K o -schemeis special. We denote by ( SpF/K o ) the category of special formal K o -schemes.There is a functor ( · ) : ( SpF/K o ) → ( sf t/ e K ) : X X to the category ( sf t/ e K ) of separated e K -schemes of finite type, where X is the reduction of X (the closed subscheme of X defined by the largest ideal of definitionin O X ).Moreover, we consider the functor( · ) s : ( SpF/K o ) → ( SpF/ e K ) : X X s = X × Spf K o Spec e K In fact, if | · | L is non-trivial, sp X can be enhanced to a morphism of ringed sites: thespecialization morphism sp X : X η → X , where X η is endowed with the strong G -topology; notethat | X | = | X s | . JOHANNES NICAISE mapping X to its special fiber X s . If X is stf t over K o , then X s is a scheme,and X is the maximal reduced closed subscheme of X s ; in any case, ( X s ) ∼ = X .For any special formal K o -scheme X and any value r ∈ ]0 , X ( r )the formal scheme X viewed as a special formal K or -scheme via the identification K o = K or . We denote by X ( r ) η the generic fiber of X ( r ) in the category of K r -analytic spaces. We also put X (0) = X s . It is a special formal scheme over e K , andwe denote its generic fiber in the category of K -analytic spaces by X (0) η . For each r ∈ [0 , sp X ( r ) : X ( r ) η → X .If X is a separated e K -scheme of finite type, then we can view X also as a stf t formal e K -scheme, and there exists a canonical morphism of K -analytic spaces X η → X an , which is an isomorphism iff X is proper [27, 1.10]. Hence, if X is aproper stf t formal R -scheme, then X (0) η is canonically isomorphic to X ans .A stf t formal K o -scheme is called strictly semi-stable if it can be covered withaffine open formal subschemes U , endowed with an ´etale morphism of formal K o -schemes of the form U → Spf K o { x , . . . , x m } / ( x · x · . . . · x p − π )for some p ≤ m , where π generates the maximal ideal of K o (in particular, π = 0if the valuation on K is trivial).This definition includes as a special case the class of strictly semi-stable e K -varieties, i.e. varieties which admit Zariski-locally an ´etale map to a e K -scheme ofthe form Spec e K [ x , . . . , x m ] / ( x · x · . . . · x p ). If X is a strictly semi-stable formal K o -scheme, then X s is a strictly semi-stable e K -variety.If X is a strictly semi-stable e K -variety, we denote by Irr ( X ) its set of irreduciblecomponents. For any non-empty subset J of Irr ( X ), we put X J = ∩ V ∈ J V and X oJ = X J \ ∪ W ∈ ( Irr ( X ) \ J ) W .2.3. The analytic Milnor fiber.
Let k be any field, and put K = k (( t )), endowedwith the t -adic valuation. Fix a value r ∈ ]0 , X be a variety over k , endowedwith a morphism of k -schemes f : X → Spec k [ t ]. The t -adic completion of f isa stf t formal K o -scheme X , whose special fiber X s is canonically isomorphic tothe fiber of f over the origin. For any closed point x of X s , the set sp − X ( r ) ( x )is open in X ( r ) η and inherits the structure of a K r -analytic space. By [9, 0.2.7]it is canonically isomorphic to the generic fiber of the special formal K or -schemeSpf b O X,x , where the K or -structure is defined by f . In [21], we called sp − X ( r ) ( x ) the analytic Milnor fiber of f at x , and we denoted it by F x . By [6, 3.5], if k is separablyclosed, the ℓ -adic cohomology of F x is canonically isomorphic to the cohomology ofthe stalk of the complex of ℓ -adic nearby cycles at x (here ℓ is a prime invertible in k ): for each integer i ≥
0, there is a canonical G ( K s /K )-equivariant isomorphism H i ´ et ( F x b × K r c K sr , Q ℓ ) ∼ = R i ψ η ( Q ℓ ) x The simplicial set associated to a strictly semi-stable variety
Definitions.
In this section, we define the simplicial complex ∆( X ) associatedto a strictly semi-stable e K -variety X , and we list some basic properties. Remark.
In [27, § X ) as an oriented simplicial complexby choosing a total order on the set of irreducible components of X . It seemsmore natural to construct ∆( X ) as an unoriented simplicial set, independent of INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 5 all choices. This is the approach we adopt here. The result is isomorphic to theunoriented simplicial set underlying Thuillier’s construction. (cid:3)
For any pair of categories C , D , we denote by D o C the category of presheaveson D with values in C . Its objects are functors D o → C , where D o denotes theopposite category of D , and its morphisms are natural transformations of functors.We denote by ( Ens ) the category of sets.For any integer n ≥
0, we denote by [ n ] the set { , . . . , n } , and we define acategory ∆ whose objects are the sets [ n ] for n ≥
0, and whose arrows are maps ofsets. The category ∆ o ( Ens ) is called the category of (unoriented) simplicial sets.The object of ∆ o ( Ens ) represented by [ n ] is called the standard n -simplex, anddenoted by ∆[ n ]. If Σ is a simplicial set and n ≥ n ])the set of n -simplices of Σ. Note that there exists a natural bijection Σ([ n ]) ∼ = Hom ∆ o ( Ens ) (∆[ n ] , Σ). A n -simplex γ is called degenerate if there exists a map ofsets g : [ n ] → [ n −
1] such that γ belongs to the image of Σ( g ) : Σ([ n − → Σ([ n ]). A m -simplex γ and a n -simplex γ are called equivalent if there exist maps [ m ] → [ n ]and [ n ] → [ m ] mapping γ to γ , resp. γ to γ . For any set I , we define theassociated simplicial set ∆ I by∆ I ([ n ]) = Hom ( Ens ) ([ n ] , I )for all n ≥ | · | : ∆ o ( Ens ) → ( Ke ) : Σ
7→ | Σ | where ( Ke ) denotes the category of Kelley spaces (topological Hausdorff spaces T such that a subset of T is closed iff its intersection with all compact subsets of T is closed). This functor is characterized by the fact that it commutes with directlimits, maps the standard n -simplex ∆[ n ] to the topological n -simplex∆ n = { ( u , . . . , u n ) ∈ [0 , n +1 | n X i =0 u i = 1 } and sends a map of sets α : [ m ] → [ n ] to the unique affine map | α | : ∆ m → ∆ n sending the vertex v i of ∆ m to the vertex v α ( i ) of ∆ n (for n ≥ i ∈ [ n ], thevertex v i of ∆ n is the point with coordinates u j = δ ij , j = 0 , . . . , n ).If Σ is a simplicial set, then a n -cell of | Σ | is the image of the interior (∆ n ) o underthe map | γ | : ∆ n → | Σ | induced by some non-degenerate n -simplex γ ∈ Σ([ n ]).Two simplices γ ∈ Σ([ m ]) and γ ∈ Σ([ n ]) are equivalent, iff the images of thecorresponding maps | γ | : ∆ m → | Σ | and | γ | : ∆ n → | Σ | coincide.Now let X be a strictly semi-stable e K -variety. We denote by Str ( X ) the setconsisting of the generic points of the closed subsets X J of X , where J varies overthe non-empty subsets of Irr ( X ). In particular, Str ( X J ) ⊂ Str ( X ) is the set ofgeneric points of the smooth variety X J . Identifying an element of Irr ( X ) with itsgeneric point, we can consider Irr ( X ) as a subset of Str ( X ) in a natural way.For any point x of X , we denote by Ψ( x ) the set of elements of Irr ( X ) containing x ; if we want to make X explicit, we write Ψ X ( x ) instead of Ψ( x ). For any point x of Str ( X ), we denote by S x the connected component of X o Ψ( x ) containing x . Then x is the generic point of S x , and { S x | x ∈ Str ( X ) } is a finite stratification of X into smooth irreducible locally closed subsets. We call any union of strata a stratasubset of X . JOHANNES NICAISE
We define a partial ordering on
Str ( X ) as follows: x ≤ y iff y belongs to theZariski closure of { x } in X . For each x ∈ Str ( X ) we consider the simplicial set∆( x ) = ∆ Ψ( x ) . For x ≤ y , we have Ψ( x ) ⊂ Ψ( y ), and hence a natural morphism ofsimplicial sets ∆( x ) → ∆( y ). This defines a functor∆( · ) : Str ( X ) → ∆ o ( Ens )and we define ∆( X ) as its colimit:∆( X ) = lim −→ Str ( X ) ∆( · )We can give a more explicit construction of ∆( X ) as follows. Denote, for each n ≥
0, by D ( X )([ n ]) the set of couples ( x, f ) with x ∈ Str ( X ), and f a surjection[ n ] → Ψ( x ). For any map α : [ m ] → [ n ], we define a map D ( X )( α ) : D ( X )([ n ]) → D ( X )([ m ]) : ( x, f ) ( x ′ , f ′ )where f ′ := f ◦ α , and x ′ is the unique point of Str ( X ) such that Ψ( x ′ ) = Im ( f ′ )and x belongs to the Zariski closure of { x ′ } in X . In this way, D ( X ) becomes asimplicial set. The following lemma can be verified in a straightforward way. Lemma 3.1.
There exists a canonical isomorphism of simplicial sets ∆( X ) ∼ = D ( X ) . In particular, for each n ≥ , there is a canonical bijection between theset of equivalence classes of non-degenerate n -simplices (or, equivalently, the set of n -cells) of ∆( X ) and the set ∪ J ⊂ Irr ( X ) , | J | = n +1 Str ( X J ) . Hence, the non-degenerate n -simplices γ of ∆( X ) correspond to couples ( x, f )with x ∈ Str ( X ) and f a bijection [ n ] ∼ = Ψ( X ). A point z of the correspondingcell of | ∆( X ) | is the image of a unique point ( u , . . . , u n ) of (∆ n ) o under the map | γ | : ∆ n → | ∆( X ) | , and we define a tuple v ∈ ]0 , Ψ( x ) by v ( i ) = u f − ( i ) for i ∈ Ψ( x ). This tuple is invariant under equivalence of non-degenerate n -simplices,and the point z of | ∆( X ) | is completely determined by the couple ( x, v ). Definition 3.2 (Barycentric representation) . We call v the tuple of barycentriccoordinates of the point z , and we call ( x, v ) the barycentric representation of z . By Lemma 3.1, there is a natural bijection x C x from Str ( X ) to the set ofcells of | ∆( X ) | . Whenever E is a strata subset of X , we denote by | ∆ E ( X ) | theunion of the cells C x with x ∈ Str ( X ) ∩ E .3.2. Comparison with Berkovich’ definition.
Denote by e ∆ the subcategory of∆ with the same objects but with only injective maps. This is a full subcategoryof the category Λ considered in [7, p. 24]. If X is a strictly semi-stable e K -variety,then Berkovich defines in [7, p. 29] an object C ( X ) of the category Λ o ( Ens ) ofpresheaves on Λ, and its geometric realization | C ( X ) | . The aim of this section is tocompare these objects with the simplicial set ∆( X ) defined above, and its geometricrealization | ∆( X ) | .A simplicial set Σ is called non-degenerate, if for any injective map [ m ] → [ n ]in ∆, the induced map Σ([ n ]) → Σ([ m ]) takes non-degenerate n -simplices to non-degenerate m -simplices. A morphism of simplicial sets is called non-degenerate ifit takes non-degenerate simplices to non-degenerate ones. We denote the subcate-gory of ∆ o ( Ens ) consisting of non-degenerate simplicial sets with non-degeneratemorphisms between them by ∆ o ( Ens ) nd . INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 7
The embedding of e ∆ in Λ, resp. ∆ induces forgetful functors F : Λ o ( Ens ) → ( e ∆) o ( Ens ) and G : ∆ o ( Ens ) → ( e ∆) o ( Ens ). Lemma 3.3.
The functor G : ∆ o ( Ens ) → ( e ∆) o ( Ens ) has a left adjoint H :( e ∆) o ( Ens ) → ∆ o ( Ens ) . This functor is a faithful embedding, its image is containedin ∆ o ( Ens ) nd , and H : ( e ∆) o ( Ens ) → ∆ o ( Ens ) nd is an equivalence of categories.Proof. For any integer n ≥
0, we denote by e ∆[ n ] the object of ( e ∆) o ( Ens ) repre-sented by [ n ]. For any object S of ( e ∆) o ( Ens ), we denote by e ∆ /S the categoryof morphisms α : e ∆[ n ] → S , with n ≥
0, where a morphism in e ∆ /S from α to β : e ∆[ m ] → S is a map f : e ∆[ n ] → e ∆[ m ] in ( e ∆) o ( Ens ) with α = β ◦ f . We definethe functor H by putting H S ( α ) = ∆[ n ] for any object α : e ∆[ n ] → S of e ∆ /S , and H ( S ) = lim −→ e ∆ /S H S ( · )In particular, H ( e ∆[ n ]) = ∆[ n ]. The action of H on morphisms in ( e ∆) o ( Ens ) is theobvious one.Now we check that H is indeed a left adjoint for G . Let Σ be any object of∆ o ( Ens ), and let S be any object of ( e ∆) o ( Ens ). By definition,
Hom ∆ o ( Ens ) ( H ( S ) , Σ) = lim −→ e ∆[ n ] → S Σ([ n ])and since Σ([ n ]) = G (Σ)([ n ]) = Hom ( e ∆) o ( Ens ) ( e ∆[ n ] , G (Σ))it suffices to note that S ∼ = lim −→ e ∆[ n ] → S e ∆[ n ]for any object S of ( e ∆) o ( Ens ), by [15, II.1.1].For any object S of ( e ∆) o ( Ens ), and each integer n ≥
0, we consider the set S n ( S ) consisting of triples ( p, f, γ ) where p ≥ f is a surjection [ n ] → [ p ], and γ is an element of S ([ p ]). We define an equivalence relation on S n ( S ) by stipulatingthat ( p, f, γ ) ∼ ( p ′ , f ′ , γ ′ ) iff p = p ′ and there exists an automorphism ϕ of [ p ] suchthat f ′ = ϕ ◦ f and γ = S ( ϕ )( γ ′ ).It is not hard to see that, for each integer n ≥
0, there exists a canonical bijectionbetween the set of n -simplices H ( S )([ n ]) and the quotient set S n ( S ) / ∼ . The n -simplex represented by ( p, f, γ ) is non-degenerate iff p = n . If α : [ m ] → [ n ] is amorphism in ∆, then H ( S )( α ) : H ( S )([ n ]) → H ( S )([ m ])maps ( p, f, γ ) to ( q, ( f ◦ α : [ m ] → Im ( f ◦ α ) ∼ = [ q ]) , γ ′ )where we chose an isomorphism [ q ] ∼ = Im ( f ◦ α ) and γ ′ is the image of γ in H ( S )([ q ])w.r.t. the inclusion map [ q ] ∼ = Im ( f ◦ α ) → [ p ]. Using this description, one can seethat for any morphism h : S → T in ( e ∆) o ( Ens ), the image H ( h ) : H ( S ) → H ( T )is a non-degenerate morphism between non-degenerate simplicial sets.To conclude, we construct a quasi-inverse I for H : ( e ∆) o ( Ens ) → ∆ o ( Ens ) nd .For any non-degenerate simplicial set Σ and any n ≥
0, we define I (Σ)([ n ]) asthe set of non-degenerate n -simplices of Σ. By the definitions of non-degenerate JOHANNES NICAISE simplicial set and non-degenerate morphism, we can make I into a functor in theobvious way. It is easy to see that I is quasi-inverse to H . (cid:3) Proposition 3.4.
For any strictly semi-stable e K -variety X , there exists a canonicalisomorphism of simplicial sets α : ( H ◦ F )( C ( X )) → ∆( X ) . In particular, ∆( X ) is non-degenerate. Moreover, there exists a canonical homeomorphism | ∆( X ) | →| C ( X ) | .Proof. Recall that, for any point x of Str ( X ), we denote by Ψ( x ) the set of ir-reducible components of X containing x . Then, by definition, for any integer n ≥ F ( C ( X ))([ n ]) is the set of pairs ( x, α ) with x ∈ Str ( X ) and α a bijection[ n ] → Ψ( x ). By the construction of the functor H in the proof of Lemma 3.3, we seethat ( H ◦ F )( C ( X ))([ n ]) is the set of pairs ( x, β ) with x ∈ Str ( X ) and β a surjection[ n ] → Ψ( x ). Now the existence of the canonical isomorphism follows from Lemma3.1. The existence of a canonical homeomorphism | ( H ◦ F )( C ( X )) | → | C ( X ) | isclear from Berkovich’ construction of | C ( X ) | . (cid:3) The skeleton of a strictly semi-stable formal scheme.
Let ( L, | · | L )be an arbitrary non-archimedean field, with ring of integers L o and residue field e L (we do not exclude the trivial absolute value). We recall a particular case ofBerkovich’ definition of the skeleton S ( X ) of a poly-stable formal L o -scheme X ,and his construction of a strong deformation retract of X η onto S ( X ) (see [7, § Case 1.
Suppose that X = Spf A with A = L o { x , . . . , x m } / ( x · . . . · x p − α )for some p ≤ m , and with α ∈ L o , | α | L <
1. Each element of A L := A ⊗ L o L has aunique representant in the set D = { X i ∈ N [ m ] a i x i ∈ L { x , . . . , x m } | a i = 0 if min { i , . . . , i p } > } i.e. the natural map D → A L is a bijection (and even an isometry if we endow A L with the quotient norm w.r.t. the given presentation L { x , . . . , x m } → A L ). Put S = { r ∈ [0 , [ p ] | r · . . . · r p = | α | L } and consider the map θ : S → X η mapping r to the element θ ( r ) of X η = M ( A L )defined by θ ( r ) : D → R + : X i a i x i max i ∈ N [ m ] {| a i | L r i } where r i = r i . . . . .r i p p with the convention that 0 = 1. The map θ is a homeo-morphism onto its image θ ( S ), which is by definition the skeleton S ( X ) of X , and θ is right inverse to the map φ : X η → S mapping a point z of X η to the tuple( | x ( z ) | , . . . , | x p ( z ) | ). If we put τ X = θ ◦ φ , then Berkovich constructed a naturalstrong deformation retract Φ X : X η × [0 , → X η with Φ X ( · ,
1) = τ X . Case 2.
Now we consider the case where X admits an ´etale map h : X → Y =Spf A , with A as above. Then the skeleton S ( X ) is the inverse image of S ( Y )under h η , and it does not depend on the choice of the map h . Moreover, Berkovichdescribes the map Φ Y in terms of a certain torus action, which lifts uniquely to X η ;in this way, he defines a natural strong deformation retract Φ X : X η × [0 , → X η of X η onto S ( X ), such that Φ Y ( h η ( x ) , ρ ) = h η ◦ Φ X ( x, ρ ) for each point x of X η and each ρ ∈ [0 , X does not depend on h . INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 9
Case 3.
Finally, if X is a stf t formal L o -scheme that can be covered by open for-mal subschemes with the property of Case 2, then the construction of the skeleton S ( X ) and the strong deformation retract Φ X are obtained by gluing the construc-tions in the previous step. We put τ X = Φ X ( · ,
1) : X η → S ( X ). In particular,Case 3 applies to all strictly semi-stable formal K o -schemes.The map Φ X has the following property: for any point x of X η , we have ( sp X ◦ Φ X )( x, ρ ) = sp X ( x ) for ρ ∈ [0 , sp X ◦ τ X )( x ) is the generic point of thestratum of the strictly semi-stable e L -variety X s containing sp X ( x ). In particular,if E is a strata subset of X s , then Φ X restricts to a strong deformation retract sp − X ( E ) × [0 , → sp − X ( E )onto S ( X ) ∩ sp − X ( E ).Berkovich constructed a natural homeomorphism S ( X ) ∼ = | ∆( X s ) | (here we usethe canonical homeomorphism | ∆( X s ) | ∼ = | C ( X s ) | established in Proposition 3.4).If E is a strata subset of X s , then this homeomorphism identifies S ( X ) ∩ sp − X ( E )with | ∆ E ( X s ) | (cf. [7, 5.4]).We will give an explicit description of the composed map τ X ( r ) : X ( r ) η → S ( X ( r )) ∼ = | ∆( X s ) | if X is a strictly semi-stable formal K o -scheme, and r ∈ [0 , n ≥ q ∈ [0 , nq = { ( u , . . . , u n ) ∈ [0 , [ n ] | Y i ∈ [ n ] u i = q } In [7, § α : ∆ n → Σ nq . Itidentifies the face of ∆ n corresponding to a non-empty subset S of [ n ], with thesubspace of Σ nq defined by u i = 1 for i / ∈ S (note that this subspace is homeomorphicto Σ | S |− q ). If w is a point of ∆ n , we call the tuple α ( w ) in Σ nq the q -colouredcoordinates of w .For any non-empty finite set I and any q ∈ [0 , α induces a map α : { u ∈ [0 , I | X i ∈ I u ( i ) = 1 } → { u ∈ [0 , I | Y i ∈ I u ( i ) = q } by choosing a bijection I ∼ = [ n ] for some n ≥
0; the map is independent of thischoice.
Definition 3.5 ( q -coloured representation) . We fix a value q ∈ [0 , . Let X bea strictly semi-stable e K -variety, and let z be a point of | ∆( X ) | with barycentricrepresentation ( x, v ) (see Definition 3.2). We define the q -coloured coordinates of z as the image of v under the map α : { u ∈ [0 , Ψ( x ) | X i ∈ I u ( i ) = 1 } → { u ∈ [0 , Ψ( x ) | Y i ∈ I u ( i ) = q } and we call ( x, α ( v )) the q -coloured representation of z . Lemma 3.6.
Let X be a strictly semi-stable formal K o -scheme, and fix a value r ∈ [0 , . Let z be any point of the special fiber X s , and let x be the unique pointof Str ( X s ) such that z is contained in the stratum S x .For each element C of Ψ( x ) , we choose a generator T C of the kernel of the naturalmorphism O X ,z → O C,z . Then the image of a point z ∈ sp − X ( r ) ( z ) under the retraction τ X ( r ) : X ( r ) η → S ( X ( r )) ∼ = | ∆( X s ) | is the point with | π | K r -colouredrepresentation ( x, ( | T i ( z ) | ) i ∈ Ψ( x ) ) . Recall that π is a generator of the maximal ideal of K o . Proof. If X is of the form Spf A with A = K o { x , . . . , x m } / ( x . . . . .x p − π ), thenthis follows immediately from the constructions in [7, § X admits an ´etale morphism h : X → Y = Spf A . Shrinking X around z , wemay assume that x is the unique maximal element of Str ( X s ) (w.r.t. the partialorder defined in Section 3.1), and that h induces a bijection Irr ( X s ) ∼ = Irr ( Y s ). Inthis case, h induces isomorphisms β : ∆( X s ) ∼ = ∆( Y s ) = ∆[ p ] and γ : S ( X ( r )) ∼ = S ( Y ( r )), by Step 6 in the proof of Theorems 5.2-4 in [7].Since | ϕ ( z ) | = 1 for any unit ϕ on X , the value | T i ( z ) | only depends on i and z and not on the choice of the generator T i . Hence, we might as well take T i = h ∗ x i for i = 0 , . . . , p (we used the bijection Irr ( X s ) ∼ = Irr ( Y s ) to identify Ψ( x ) with { , . . . , p } ). Therefore, it only remains to observe that the diagram | ∆( X s ) | −−−−→ S ( X ( r )) β y y γ | ∆( Y s ) | −−−−→ S ( Y ( r ))commutes, where the horizontal arrows are the natural homeomorphisms con-structed by Berkovich (in fact, this is the definition of the upper horizontal arrowin Step 6 of Berkovich’ proof [7, p. 48]). (cid:3) Restriction to irreducible components.
Let X be a strictly semi-stable e K -variety, and let E be a union of irreducible components of X . Of course, E itself is again a strictly semi-stable e K -variety, and hence, it defines an associatedsimplicial set ∆( E ) with geometric realization | ∆( E ) | . In general, | ∆( E ) | is nothomeomorphic to | ∆ E ( X ) | . For instance, when X = Spec e K [ x, y ] / ( xy ) and E isthe component x = 0, then | ∆( E ) | is a point while | ∆ E ( X ) | is homeomorphic to thesemi-open interval [0 , Str ( E ) ⊂ Str ( X ) induces an injective morphism of simplicial sets ∆( E ) → ∆( X )and a natural continuous injection | ∆( E ) | → | ∆ E ( X ) | which is a homeomorphismonto its image.Denote by E the formal completion of X along E ; this is a special formal e K -scheme. The closed immersion of special formal e K -schemes h : E → E induces amorphism of K -analytic spaces h η : E η → E η . Proposition 3.7. If X is a strictly semi-stable e K -variety, and E is a union ofirreducible components of X , then there exists a strong deformation retract Φ XE : | ∆ E ( X ) | × [0 , → | ∆ E ( X ) | of | ∆ E ( X ) | onto | ∆( E ) | such that the diagram E η h η −−−−→ E η y τ E y τ X | ∆( E ) | Φ XE ( · , ←−−−−− | ∆ E ( X ) | commutes. INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 11
Proof.
We may assume E = X . Choose a sequence Irr ( E ) = I (0) ⊂ I (1) ⊂ . . . ⊂ I ( q + 1) = Irr ( X )such that | I ( i + 1) | = | I ( i ) | + 1 for i = 0 , . . . , q , and put E ( i ) = ∪ V ∈ I ( i ) V . Denote by E ( i ) the formal completion of E ( i ) along E , and by g ( i ) : E ( i ) → E ( i +1) the naturalclosed immersion. It suffices to construct a strong deformation retract Φ XE ( q ) of | ∆ E ( X ) | onto | ∆ E ( E ( q ) ) | such that the diagram(3.1) E ( q ) η g ( q ) η −−−−→ E η y τ E ( q ) y τ X | ∆ E ( E ( q ) ) | Φ XE ( q ) ( ., ←−−−−−− | ∆ E ( X ) | commutes: iterating the construction, we get strong deformation retracts Φ E ( i +1) E ( i ) for i = 0 , . . . , q , and these can be glued to obtain Φ XE .To simplify notation, we will denote E ( q ) by Y , E ( q ) by Y , and g ( q ) by g . Denotethe unique element in Irr ( X ) \ Irr ( Y ) by a .For any point z of | ∆( X ) | , we will denote its 0-coloured representation by( x z , u z ). The point z belongs to | ∆ E ( X ) | (resp. | ∆ E ( Y ) | ) iff Ψ X ( x z ) ∩ Irr ( E ) = ∅ (resp. iff Ψ X ( x z ) ∩ Irr ( E ) = ∅ and Ψ X ( x z ) ⊂ Irr ( Y )).For ∅ 6 = J ⊂ Irr ( X ), x a generic point of X J , and u ∈ [0 , J , we define thesupport Supp ( u ) of u as the set of indices i ∈ J with u ( i ) = 1. We identify ( x, u )with the point z ∈ | ∆( X ) | with 0-coloured representation ( x z , u z ), were u z is therestriction of u to Supp ( u ) and x z is the unique generic point of X Supp ( u ) such that x belongs to the Zariski closure of { x z } in X .Consider the functionΦ XY : | ∆ E ( X ) | × [0 , → | ∆ E ( X ) | : ( z = ( x z , u z ) , ρ ) ( x z , u ( ρ )) = Φ XY ( z, ρ )where u ( ρ ) ∈ [0 , Ψ X ( x z ) depends on x z , u z and ρ , and is defined in the followingway. Case 1. If a / ∈ Ψ X ( x z ), then u ( ρ ) = u z for each value of ρ . Case 2.
Now assume that a ∈ Ψ X ( x z ). If there exists an element b ∈ Ψ Y ( x z ) =Ψ X ( x z ) ∩ Irr ( Y ) with u z ( b ) = 0, then we put u ( ρ )( i ) = u z ( i ) for i ∈ Ψ Y ( x z ) and ρ ∈ [0 , u z ( a ) for i = a and ρ ∈ [0 , / u z ( a ) + (1 − u z ( a ))(2 ρ −
1) for i = a and ρ ∈ [1 / , Case 3.
Finally, suppose that u z ( i ) = 0 for all i ∈ Ψ Y ( x z ). Then necessarily u z ( a ) = 0. We put u ( ρ )( a ) = 0 for ρ ∈ [0 , / u ( ρ )( i ) = u z ( i ) − ρ · min j ∈ Ψ X ( x z ) ( u z ( j ))for i ∈ Ψ Y ( x z ) and ρ ∈ [0 , / b ∈ Ψ Y ( x z ) with u (1 / b ) = 0, so that the definition in Case 2 applies to v := u (1 / ∈ [0 , Ψ X ( x z ) ,and we put u ( ρ ) = v ( ρ ) for ρ ∈ [1 / , XY is continuous, and defines a strong deformationretract of | ∆ E ( X ) | onto | ∆ E ( Y ) | . Let us check the commutativity of diagram (3.1).Let z be any point of Y η , and put sp Y ( z ) = sp X ( z ) = z . For each element C of Ψ X ( z ), we choose a generator T C of the kernel of the natural morphism O X,z → O C,z . Denote by y and x the points of Str ( Y ), resp. Str ( X ) such that z belongs to the corresponding stratum of Y , resp. X . Then by Lemma 3.6 wehave to show that Φ XY ( · ,
1) takes the point z ′ = ( τ X ◦ g η )( z ) of | ∆ E ( X ) | with 0-coloured representation ( x, ( | T i ( z ) | ) i ∈ Ψ X ( z ) ) to the point τ Y ( z ) of | ∆ E ( Y ) | with0-coloured representation ( y, ( | T i ( z ) | ) i ∈ Ψ Y ( z ) ).If Ψ X ( z ) = Ψ Y ( z ) this is obvious, so assume that Ψ X ( z ) = Ψ Y ( z ) ⊔{ a } . Since z is an element of Y η , there exists an element i of Ψ Y ( z ) with | T i ( z ) | = 0. ByCase 2 of the definition, Φ XY ( z ′ ,
1) is given by the couple ( x, v ) with v ∈ [0 , Ψ X ( z ) , v ( i ) = | T i ( z ) | for i = a , and v ( a ) = 1. By the identifications we made, this isexactly the point τ Y ( z ) (since y is the unique generic point of X Ψ Y ( z ) such that x belongs to the Zariski closure of { y } in Y ). (cid:3) Corollary 3.8.
The map h η : E η → E η is a homotopy equivalence.Proof. By Proposition 3.7, Φ XE ( · ,
1) is a homotopy equivalence with as homotopyinverse the natural embedding i : | ∆( E ) | → | ∆ E ( X ) | . Moreover, τ E and τ X are homotopy equivalences with as homotopy inverses the natural embeddings σ E : | ∆( E ) | → E η , resp. σ X : | ∆ E ( X ) | → E η (we used Berkovich’ naturalhomeomorphisms to identify | ∆( E ) | with the skeleton S ( E ) and | ∆ E ( X ) | with S ( X ) ∩ sp − X ( E )). Now the fact that h η is a homotopy equivalence follows fromthe commutativity of the diagram in the statement of Proposition 3.7, and thecommutativity of the diagram | ∆( E ) | i −−−−→ | ∆ E ( X ) | σ E y y σ X E η h η −−−−→ E η which follows easily from the description of the skeleton in Section 3.3. (cid:3) Homotopy type of the analytic Milnor fiber
Let k be any field, and put R = k [[ t ]] and K = k (( t )), endowed with the t -adicvaluation (so R = K o ). For each 0 < r <
1, we denote by | · | r the t -adic absolutevalue on K with | t | r = r . We fix an algebraic closure K a of K .We endow k with its trivial absolute value |·| , and we put R := Spf R . Moreover,we endow k [ t ] with the trivial Banach norm (this norm coincides with the Gaussnorm if we view k [ t ] as the algebra of convergent power series k { t } ). The formalscheme R is a special formal k -scheme in the sense of [6]. We denote its genericfiber by R η ; it coincides with the open unit disc D (1) = { x ∈ M ( k [ t ]) | | t ( x ) | < } .The following result was stated in Step 3 of the proof of [3, Thm 4.1], withoutproof. We include the elementary proof for the reader’s convenience. Lemma 4.1.
The natural map
Ψ : D (1) → [0 ,
1[ : x
7→ | t ( x ) | is a homeomorphism. If we put p r := Ψ − ( r ) for ≤ r < , then the residue field H ( p r ) is K = ( k, | · | ) for r = 0 , and K r = ( K, | · | r ) for < r < .Proof. The map Ψ is obviously continuous. Its inverse isΨ − : [0 , → D (1) : r p r INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 13 where p r is bounded multiplicative semi-norm in M ( k [ t ]) defined by p r ( P ni =0 a i t i ) :=max i | a i | r i (with the convention that 0 = 1).Indeed: it is clear that Ψ ◦ Ψ − is the identity. Now we show that Ψ − is alsoleft inverse to Ψ. Choose x in D (1) and put r = | t ( x ) | . A classical trick showsthat the residue field H ( x ) of x is ultrametric: for f, g in k [ t ], and n ≥
0, we have | ( f + g ) n ( x ) | = | ( n X i =0 (cid:18) ni (cid:19) f i g n − i )( x ) | ≤ n X i =0 | (cid:18) ni (cid:19) | | f ( x ) | i | g ( x ) | n − i ≤ ( n + 1)(max { | f ( x ) | , | g ( x ) | } ) n and taking n -th roots and sending n to ∞ , we see that | ( f + g )( x ) | ≤ max { | f ( x ) | , | g ( x ) | } .If r = 0, then the ultrametric property implies x = p , so we may assume0 < r <
1. In this case, | at i ( x ) | 6 = | a ′ t j ( x ) | for a, a ′ ∈ k ∗ and i = j , so we get | ( P ni =0 a i t i )( x ) | = max i | a i | r i .It remains to prove that Ψ − is continuous. By definition of the spectral topology,it suffices to show that [0 , → R + : r
7→ | f ( p r ) | = p r ( f ) is continuous for each f ∈ k [ t ]. This, however, is clear.Finally, we determine the residue fields of the points p r . Since | P ni =0 a i t i ( x ) | = | a | , we see that H ( p ) = ( k, | · | ). For r >
1, no element of k [ t ] \ { } vanishes in p r , so H ( p r ) is the completion of k ( t ) w.r.t. p r ; this is exactly ( K, | · | r ). (cid:3) If X is a special formal R -scheme, then we can also consider X as a specialformal k -scheme via the composition X → R → Spec k . We denote this object by X k . This yields a forgetful functor( SpF/R ) → ( SpF/k ) : X X k from the category ( SpF/R ) of special formal R -schemes to the category ( SpF/k )of special formal k -schemes. We can associate to X k its generic fiber X kη (a K -analytic space), and there is a natural specialization map sp X k : X kη → X .The map of special formal k -schemes h : X k → R induces a map of K -analyticspaces h η : X kη → R η . Its fibers can be described as follows. Lemma 4.2.
For ≤ r < , there is a canonical isomorphism X ( r ) ∼ = X k b × R Spf H ( p r ) o and the fiber of h η over p r is canonically isomorphic to the K r -analytic space X ( r ) η .Proof. By the construction of the generic fiber in [6, § X = Spf A , with A of the form R [[ x , . . . , x m ]] { y , . . . , y n } . Then X kη is the polydisc { z ∈ M ( k [ t, x , . . . , y n ]) | | t ( z ) | < | x i ( z ) | < i = 1 , . . . , m } where k [ t, x , . . . , y n ] carries the trivial Banach norm, and the map h η sends thebounded multiplicative semi-norm z to its restriction to k [ t ]. Now we observe that k [ t, x , . . . , y n ] b ⊗ k [ t ] H ( p ) ∼ = k [ x , . . . , y n ]while k [ t, x , . . . , y n ] b ⊗ k [ t ] H ( p r ) ∼ = K r { x , . . . , y n } for 0 < r < (cid:3) Proposition 4.3.
Let X be a special formal R -scheme. If we denote by λ the map | X kη | → | R η | ∼ = [0 , , then for each r ∈ [0 , , there exists a canonical homeomor-phism | X ( r ) η | ∼ = λ − ( r ) such that the square | X ( r ) η | ∼ −−−−→ λ − ( r ) ⊂ | X kη | sp X ( r ) y y sp X k | X | | X | commutes.Moreover, for any < r < , there exists a canonical homeomorphism φ : λ − ( ]0 ,
1[ ) → | X ( r ) η |× ]0 , such that the composition of φ with the projection | X ( r ) η |× ]0 , → ]0 , coincideswith λ : λ − ( ]0 ,
1[ ) → ]0 , .Proof. It suffices to consider the case where X = Spf A is affine, with A of theform A = R [[ x , . . . , x m ]] { y , . . . , y n } / ( f , . . . , f ℓ )Then X kη is a closed subset of the polydisc E := { z ∈ M ( k [ t, x , . . . , y n ]) | | t ( z ) | < | x i ( z ) | < i = 1 , . . . , m } where k [ t, x , . . . , y n ] carries the trivial Banach norm, and this closed subset isdefined by the equations z ( f j ) = 0, j = 1 , . . . , ℓ . The map λ sends z to the point | t ( z ) | of | R η | ∼ = [0 , λ − ( r ) is canonically homeomorphic to | X ( r ) η | for each r , andthat this homeomorphism is compatible with the specialization maps sp X ( r ) and sp X k : w.r.t. both of these maps, the image of a point z in λ − ( r ) is the open primeideal { f ∈ A | | f ( z ) | < } of A .Now fix r in ]0 , λ − ( ]0 ,
1[ ) → | X ( r ) η |× ]0 , x ( x log λ ( x ) r , λ ( x ))where we denote by x log λ ( x ) r the bounded multiplicative semi-norm in M ( k [ t, x , . . . , y n ])sending f ∈ k [ t, x , . . . , y n ] to x ( f ) log λ ( x ) r ; then clearly λ ( x log λ ( x ) r ) = r . The mapΨ is a bijection, with inverseΨ − : | X ( r ) η |× ]0 , → λ − ( ]0 ,
1[ ) : ( x, ρ ) x log r ρ and one checks immediately that both Ψ and Ψ − are continuous. (cid:3) Proposition 4.4.
Let X be a strictly semi-stable formal R -scheme, let E be aunion of irreducible components of X s , and denote by E the formal completion of X along E .(1) The inclusion map E ( r ) η → E kη is a homotopy equivalence, for each r ∈ [0 , .(2) The natural map E η → E (0) η , induced by the closed immersion E → E (0) ,is a homotopy equivalence. In particular, if E is proper, then E an → E (0) η is ahomotopy equivalence. INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 15
Proof. (1) In the last part of the proof of [3, 4.1], Berkovich constructs the so-calledskeleton S ( X / R ) ⊂ X kη associated to the morphism of special formal k -schemes X k → R : it is the union of the skeletons S ( X ( r )) ⊂ X ( r ) η ∼ = λ − ( r ), r ∈ [0 , X kη × [0 , → X kη which coincides withΦ X ( r ) on λ − ( r ) × [0 , X kη onto S ( X / R ).By [7, 5.2(iv)] Φ restricts to a strong deformation retract of ( sp X k ) − ( E ) onto S ( X / R ) ∩ ( sp X k ) − ( E ) compatible with λ (i.e. it is a strong deformation retracton each fiber of λ ).Next, Berkovich states that there exists a homeomorphism | ∆( X s ) | × R η → S ( X / R ) such that the projection of | ∆( X s ) | × R η on the second factor R η corre-sponds to the map λ on S ( X / R ). His construction contains a minor error, but itcan easily be corrected as follows. By the same arguments, it suffices to constructa homeomorphism f : Σ n × [0 , → { ( x, ρ ) ∈ R [ n ] × [0 , | x ∈ Σ nρ } for each n , such that these homeomorphisms are compatible with the face maps(so that we get good gluing properties). We can define f by sending ( y, ρ ) to( x, ρ ), where x is the unique intersection point of Σ nρ and the segment in R [ n ] joining y and (1 , . . . , S ( X / R ) ∩ ( sp k ) − ( E )with | ∆ E ( X ) | × R η . This proves (1).(2) Since X s is a strictly semi-stable k -variety, and E (0) is isomorphic to thecompletion of X s along E , this follows immediately from Corollary 3.8 and the factthat E η ∼ = E an for proper E . (cid:3) Lemma 4.5.
Assume that k is algebraically closed. If X is a strictly semi-stableformal R -scheme, and E is any strata subset of X s , then the natural map sp − X ( r ) ( E ) b × K r L → sp − X ( r ) ( E ) is a homotopy equivalence for any r ∈ ]0 , and any isometric embedding of non-archimedean fields K r ⊂ L .Proof. Put Y = X b × Spf R Spf L o . Then Y η ∼ = X ( r ) η b × K r L , Y s ∼ = X s × k e L , andthese natural isomorphisms commute with the specialization maps sp X ( r ) and sp Y ,so that they induce a natural isomorphism sp − X ( r ) ( E ) b × K r L ∼ = sp − Y ( F )where F denotes the inverse image of E in Y s .Since k is algebraically closed, the natural map h s : Y s → X s induces a bijection Irr ( Y s ) ∼ = Irr ( X s ) and a homeomorphism α : | ∆( Y s ) | ∼ = | ∆( X s ) | identifying | ∆ F ( Y s ) | with | ∆ E ( X s ) | . If we denote by h the natural morphism Y η → X ( r ) η ,it is easy to see from the description in Section 3.3 that the diagram sp − Y ( F ) τ Y −−→ | ∆( Y s ) | ∼ = −−→ S ( Y ) ∩ sp − Y ( F ) −−→ sp − Y ( F ) h y α y ∼ = ∼ = y h y h sp − X ( r ) ( E ) τ X ( r ) −−−−→ | ∆( X s ) | ∼ = −−→ S ( X ( r )) ∩ sp − X ( r ) ( E ) −−→ sp − X ( r ) ( E )commutes (the right horizontal arrows are the inclusion maps). (cid:3) Proposition 4.6.
Assume that k is an algebraically closed field of characteristiczero, and fix r ∈ ]0 , . Let X be a proper flat R -variety such that X × R K is smoothover K , and such that X s has at most one singular point x , and denote by X its t -adic completion. Then there exists a canonical long exact sequence in integralsingular cohomology . . . −→ H i (( X s ) an , Z ) −→ H i ( X ( r ) η , Z ) i ∗ −→ e H i ( ] x [ , Z ) −→ H i +1 (( X s ) an , Z ) −→ . . . with ] x [= sp − X ( r ) ( x ) , and where i : ] x [ → X ( r ) η is the inclusion map and e H ∗ ( · ) isreduced cohomology.Proof. Passing to a finite extension of R , we may assume that there exists a propermorphism of R -varieties h : Y → X such that Y is strictly semi-stable, such that h is an isomorphism over the complement of x in X , and such that E := h − ( x ) isa union of irreducible components of Y s . We denote by Y the t -adic completion of Y , and by E the formal completion of Y along E .The morphism h induces a surjective morphism of k -analytic spaces h ans : ( Y s ) an → ( X s ) an ; since X and Y are proper over R , ( Y s ) an and ( X s ) an are compact Haus-dorff spaces. Moreover, h ans maps ( Y s \ E ) an ∼ = ( Y s ) an \ E an isomorphically to( X s ) an \ { x } , and maps E an to { x } . Therefore, h ans induces a homeomorphism( Y s ) an /E an ≈ ( X s ) an , and we get a natural exact sequence . . . −→ H i (( X s ) an , Z ) −→ H i (( Y s ) an , Z ) −→ e H i ( E an , Z ) −→ H i +1 (( X s ) an , Z ) −→ . . . By Proposition 4.4 and Lemma 4.5, the natural maps X ( r ) η → Y kη , ( Y s ) an → Y kη , E an → E kη and ] x [ ∼ = E ( r ) η → E kη are all homotopy equivalences. Hence, we obtainthe desired exact sequence. (cid:3) Now we come to the main result of this section: the description of the homotopytype of the analytic Milnor fiber. First, we need an auxiliary definition. Let r ∈ ]0 , F x as a K r -analytic space. Definition 4.7 (Strictly semi-stable model) . Let X be a variety over k , endowedwith a morphism f : X → Spec k [ t ] which is flat over the origin and has smoothgeneric fiber, and let x be a closed point of the special fiber X s = f − (0) . A strictlysemi-stable model of the germ ( f, x ) of f at x consists of the following data: (1) an integer d > , and an embedding of k [ t ] -algebras A d = k [ t, u ] / ( u d − t ) → K a (2) a flat projective morphism g : Y → Spec A d whose u -adic completion isstrictly semi-stable, (3) open subschemes U and V of X , resp. Y , with x ∈ U , (4) a proper morphism ϕ : V → U × k [ t ] A d which is an isomorphism overthe complement of U s , such that g = p ◦ ϕ on V (with p the projection U × k [ t ] A d → Spec A d ) and such that ϕ − ( x ) is a union of irreduciblecomponents of the special fiber Y s of g .We’ll denote this strictly semi-stable model by ( Y, g, ϕ ) (the other data are implicitin the notation). We call d the ramification index of the strictly semi-stable model.A strictly semi-stable model of the analytic Milnor fiber F x of f at x consists ofthe following data: INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 17 (1) an integer d > , and an embedding of non-archimedean K r -fields K r ( d ) = K r [ u ] / ( u d − t ) → K ar (2) a strictly semi-stable formal K r ( d ) o -scheme Y , and a closed subvariety E of Y s which is a union of irreducible components of Y s , (3) an isomorphism of K r ( d ) -analytic spaces ϕ : sp − Y ( r ) ( E ) ∼ = F x × K r K r ( d ) .We’ll denote this strictly semi-stable model by ( Y , E, ϕ ) (the other data are implicitin the notation). We call d the ramification index of the strictly semi-stable model. It is clear that any strictly semi-stable model of ( f, x ) induces a strictly semi-stable model of F x by passing to the t -adic completion (and of course, this stillholds if we omit the projectivity condition in (2)). Proposition 4.8.
Suppose that k has characteristic zero. Let X be a variety over k , endowed with a morphism f : X → Spec k [ t ] which is flat over the origin and hassmooth generic fiber, and let x be a closed point of the special fiber X s = f − (0) .Then ( f, x ) admits a strictly semi-stable model.Proof. We may as well assume that f is projective: restrict f to an affine neighbour-hood of x , consider its projective completion, and resolve singularities at infinity.Now the result follows from the semi-stable reduction theorem [16, II]. (cid:3) Corollary 4.9.
Under the same conditions, F x admits a strictly semi-stable model. Theorem 4.10.
Suppose that k is algebraically closed (of arbitrary characteristic).Let X be a variety over k , endowed with a morphism f : X → Spec k [ t ] which isflat over the origin and has smooth generic fiber, and let x be a closed point ofthe special fiber X s = f − (0) . Suppose that F x admits a strictly semi-stable model ( Y , E, ϕ ) . Then F x is naturally homotopy-equivalent to | ∆( E ) | . In particular, thehomotopy type of | ∆( E ) | does not depend on the chosen strictly semi-stable model.Proof. Let d be the ramification index of the strictly semi-stable model ( Y , E, ϕ ).The isomorphism ϕ induces an isomorphism sp − Y ( r ) ( E ) b × K r ( d ) \ ( K r ) a ∼ = F x so the result follows from Lemma 4.5 and Proposition 4.4. (cid:3) Remark.
By the same arguments, we have the following result: suppose that k isalgebraically closed, and fix r ∈ ]0 , stf t formal R -scheme X , and assume that it admits a strictly semi-stable model h : Y → X (i.e. Y is a strictly semi-stable formal L o -scheme for some finite extension L of K r in K ar , and h is a morphism of formal R -schemes such that the induced morphism Y η → X ( r ) η × K r L is an isomorphism). Such a model exists, in particular, if k hascharacteristic zero (use embedded resolution for singularities for ( X , X s ) as in [26]and apply the algorithm for semi-stable reduction in characteristic zero [16, II]).The analytic space X ( r ) η is naturally homotopy-equivalent to | ∆( Y s ) | . In par-ticular, the homotopy type of | ∆( Y s ) | does not depend on the chosen strictly semi-stable model. This result, and the one in Theorem 4.10, are similar in nature to[27, Thm 4.8]. (cid:3) Weight zero part of the mixed Hodge structure on the nearbycohomology
Cocubical systems.
We recall the following definition. For any finite, non-empty set S , we denote by (cid:3) S the set of non-empty subsets of S , ordered by inclu-sion. For any category C , the category of S -cocubical systems in C is the categoryof covariant functors (cid:3) S → C (with natural transformations as morphisms).Let A be an abelian category, and denote by C + ( A ) the category of boundedbelow complexes in A . We fix an integer n ≥
0, and we consider a [ n ]-cocubicalsystem ( C • L ) L ∈ (cid:3) [ n ] in C + ( A ). For each object L in (cid:3) [ n ] and each p ∈ Z , we denoteby d L : C pL → C p +1 L the differential in the complex C • L ∈ C + ( A ). For each couple( L, L ′ ) of objects in (cid:3) [ n ] with L ⊂ L ′ , we denote by δ LL ′ : C • L → C • L ′ the face map which is part of the cocubical system. We define the associated simplecomplex s (cid:3) (( C • L ) L ∈ (cid:3) [ n ] ) as follows (we use the notation s (cid:3) to avoid confusion withthe simple complex s ( · ) associated to a double complex).First, we define a double complex A •• . We put A p,q = (cid:26) p, q ) ∈ Z × Z < ⊕ | L | = q +1 C pL for ( p, q ) ∈ Z × Z ≥ The horizontal differential A p,q → A p +1 ,q , for q ≥
0, is given by ⊕ | L | = q +1 { d L : C pL → C p +1 L } The restriction of the vertical differential A p,q → A p,q +1 to the component C pL isgiven by C pL → A p,q +1 : X i ∈ [ n ] \ L ( − ε ( L,i )+ p +1 δ LL ∪{ i } where ε ( L, i ) denotes the number of elements j in L with j < i .We define s (cid:3) (( C • L ) L ∈ (cid:3) [ n ] ) as the associated simple complex s ( A •• ); we’ll alsodenote it by s (cid:3) ( C • L ) to simplify notation. A map f L : C • L → D • L of [ n ]-cocubicalsystems in C + ( A ) induces a map s (cid:3) ( f L ) : s (cid:3) ( C • L ) → s (cid:3) ( D • L ) between the associ-ated simple complexes, so we get a functor s (cid:3) ( · ) from the category of [ n ]-cocubicalsystems in C + ( A ) to the category C + ( A ).Denote by C + ( A , W, F ) the category of bifiltered bounded below complexes in A (with F decreasing and W increasing). If ( C • L , W, F ) L ∈ (cid:3) [ n ] is a [ n ]-cocubicalsystem in C + ( A , W, F ) then we endow the simple complex C • = s (cid:3) ( C • L ) with thefollowing filtrations: for each n, r ∈ Z , we put F r C n = ⊕ p + q = n, q ≥ ⊕ | L | = q +1 F r C pL W r C n = ⊕ p + q = n, q ≥ ⊕ | L | = q +1 W r + q C pL We call the bifiltered complex ( s (cid:3) ( C • L ) , W, F ) the associated simple complex of thecocubical system ( C • L , W, F ) L ∈ (cid:3) [ n ] , and we put s (cid:3) ( C • L , W, F ) = ( s (cid:3) ( C • L ) , W, F )This defines a functor s (cid:3) from the category of [ n ]-cocubical systems in C + ( A , W, F )to the category C + ( A , W, F ).Let us consider some elementary examples, which will be of use later on. INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 19
Example 5.1.
Let X be a smooth complex variety and let E be a proper strictnormal crossing divisor on X , with irreducible components E i , i = 0 , . . . , n . For any L ∈ (cid:3) [ n ] , we put E L = ∩ i ∈ L E i and denote by a L : E L → X the inclusion. Considerthe bifiltered complex H L = (( a L ) ∗ Ω • E L , W, F ) where F is the stupid filtration, and W is given by W i (( a L ) ∗ Ω • E L ) = for i < a L ) ∗ Ω • E L for i ≥ In other words, H L is the image under ( a L ) ∗ of the complex component Hdg • ( E L ) C of the mixed Hodge complex of sheaves associated to the smooth and proper complexvariety E L . If L ′ is another subset of [ n ] and L ⊂ L ′ , then the closed immersion E L ′ → E L induces restriction maps H L → H L ′ , which make ( H L ) L ∈ (cid:3) [ n ] into a [ n ] -cocubical system. We denote its associated simple complex by Hdg • ( E ) C . Itis the complex component of a mixed Hodge complex of sheaves which induces thecanonical mixed Hodge structure on H ∗ ( E, Z ) [24, 3.5] . Example 5.2.
Let X be any topological space, and consider a cover { E i | i ∈ [ n ] } of X by closed subsets. For any L ∈ (cid:3) [ n ] , we put E L = ∩ i ∈ L E i and denote by a L : E L → X the inclusion.Let G • be any object in C + ( X ) (i.e. a bounded below complex of abelian sheaveson X ), and consider the [ n ] -cocubical system defined by G • L = ( a L ) ∗ ( a L ) ∗ G • . Byadjunction, we have a natural map of complexes G • → ⊕ i ∈ [ n ] ( a i ) ∗ ( a i ) ∗ G • where we wrote a i instead of a { i } . The target of this map is a direct summand of s (cid:3) ( G • L ) , so we get a map of complexes G • → s (cid:3) ( G • L ) . This is a quasi-isomorphism,since for each q ≥ , the q -th row of the double complex A •• associated to thecocubical system G • L is a resolution of G q . If f : G • → H • is a morphism in C + ( X ) , then f induces a morphism of cocubical systems f L : G • L → H • L , and thesquare G • f −−−−→ H • y y s (cid:3) ( G • L ) s (cid:3) ( f L ) −−−−→ s (cid:3) ( H • L ) commutes. Example 5.3.
We keep the notations of Example 5.2, supposing moreover that X is a smooth complex variety, and that E L is a smooth closed subvariety for each L ∈ (cid:3) [ n ] . We consider the [ n ] -cocubical system of complexes (( a L ) ∗ Ω • E L ) L ∈ (cid:3) [ n ] in C + ( X, C ) . The product of restriction maps Ω • X → ⊕ i ∈ [ n ] ( a i ) ∗ Ω • E i induces amap of complexes Ω • X → s (cid:3) (( a L ) ∗ Ω • E L ) . This map is a quasi-isomorphism: viathe quasi-isomorphisms Ω • X ∼ = C X and Ω • X ∼ = C E L and the exactness of the functor ( a L ) ∗ : C + ( E L , C ) → C + ( X, C ) , we recover the situation of Example 5.2, with G • = C X . Localized limit mixed Hodge complex.
Let S be the open complex unitdisc, and let f : X → S be a projective morphism, with X a complex manifold.We fix a complex coordinate t on S . Assume that the special fiber X s is a reducedstrict normal crossing divisor E = P i ∈ I E i . We fix a total order on I , i.e. abijection I ∼ = [ a ] for some integer a ≥
0. Let J be a non-empty subset of I , put E ( J ) = ∪ i ∈ J E i , and denote by v J : E ( J ) → X s the inclusion map. The total orderon I induces a total order on J , i.e. a bijection J ∼ = [ b ] for some integer b ≥
0. Thesetotal orders are necessary to apply the functor s (cid:3) to I - and J -cocubical systems.In [17] (see also [18]), Navarro Aznar constructed a localized limit integral mixedHodge complex of sheaves on E ( J ), whose integral component is quasi-isomorphicto v ∗ J Rψ f ( Z ) (here Rψ f is the complex analytic nearby cycle functor associated to f ). This mixed Hodge complex induces a canonical integral mixed Hodge structureon the hypercohomology spaces H i ( E ( J ) , v ∗ J Rψ f ( Z ))We will follow the approach in [23, § v ∗ J ψ Hf ( C ) of the localized limit integral mixed Hodge complexof sheaves v ∗ J ψ Hf on E ( J ), in order to fix notations (the notations we adopt herediffer slightly from the ones in [23, § L of J , we denote by I E L the defining ideal sheaf of E L = ∩ i ∈ L E i in X . Then I E L Ω • X (log E ) is a subcomplex of Ω • X (log E ), and weconsider the bifiltered complexΩ • X (log E ) | E L = Ω • X (log E ) / I E L Ω • X (log E )in C + ( E, C ), endowed with the quotient filtrations W and F (the quotient of theweight filtration, resp. of the stupid filtration on Ω • X (log E )). We define a doublecomplex A •• L by A p,qL = Ω p + q +1 X (log E ) | E L /W p (Ω p + q +1 X (log E ) | E L )for p, q ≥ A •• L as a sheaf on E ( J ). The differentials d ′ L and d ′′ L are given by d ′ L : A p,qL → A p +1 ,qL : ω ( dt/t ) ∧ ωd ′′ L : A p,qL → A p,q +1 L : ω dω We put an increasing filtration W ( M ) (the monodromy weight filtration ) on A •• L bydefining W ( M ) r A p,qL as the image of W r +2 p +1 Ω p + q +1 X (log E ) in A p,qL ; this inducesa filtration W ( M ) on the associated simple complex C • L = s ( A •• L ). We also endow C • L with a decreasing filtration F by putting F r C nL = ⊕ p + q = n,q ≥ r A p,qL .When L varies over the non-empty subsets of J , the bifiltered complexes ( C • L , W ( M ) , F )form a J -cocubical complex of bifiltered objects in C + ( E ( J ) , C ), whose associatedsimple complex s (cid:3) ( C • L , W ( M ) , F ) is denoted by v ∗ J ψ Hf ( C ). The complex s (cid:3) ( C • L ) isquasi-isomorphic to v ∗ J Rψ f ( C ).5.3. The specialization map on the mixed Hodge level.
We keep the nota-tions of Section 5.2. We would like to lift the natural specialization map C E ( J ) → v ∗ J Rψ f ( C ) to a map of bifiltered complexes Hdg • ( E ( J )) C → v ∗ J ψ Hf ( C ) INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 21 (here
Hdg • ( E ( J )) C is the complex introduced in Example 5.1). In the global case J = I , this was done in [23, § M of I , we denote by a M the closed immersion E M → E . In the proof of [23, Thm. 6.12], the following property was shown. Lemma 5.4.
There is a commutative diagram Gr Wm (Ω • X (log E )) −−−−→ ∼ = L M ⊂ I | M | = m ( a M ) ∗ Ω • E M [ − m ] y y Gr Wm (Ω • X (log E ) | E L ) ∼ = −−−−→ L M ⊂ I | M | = m ( a L ∪ M ) ∗ Ω • E L ∪ M [ − m ] for any couple non-empty subset L of I and any integer m ≥ . Here the upperhorizontal map is the isomorphism induced by the residue map, the left verticalarrow comes from the natural projection Ω • X (log E ) → Ω • X (log E ) | E L , and the rightvertical arrow is the obvious restriction map. By Example 5.3 (applied to X = E L and the cover { E L ∪ i | i ∈ I } ), we have foreach non-empty subset L of J a quasi-isomorphism( a L ) ∗ Ω • E L ∼ = ⊕ p + q = • , p ≥ ⊕ M ⊂ I, | M | = p +1 ( a L ∪ M ) ∗ Ω qE L ∪ M and using the isomorphism in Lemma 5.4, we get a natural quasi-isomorphism( a L ) ∗ Ω • E L → ⊕ p + q = • , p ≥ Gr Wp +1 (Ω p + q +1 X (log E ) | E L )whence a morphism of J -cocubical systems of bifiltered complexes(5.1) σ L : (( a L ) ∗ Ω • E L , W, F ) → ( C • L , W ( M ) , F )where the left hand side is defined as in Example 5.1. Passing to the associatedsimple complexes, we get a map of bifiltered complexes(5.2) σ : Hdg • ( E ( J )) C → v ∗ J ψ Hf ( C ) Lemma 5.5.
The square C E ( J ) spec −−−−→ v ∗ J Rψ f ( C ) y y Hdg • ( E ( J )) C σ −−−−→ v ∗ J ψ Hf ( C ) commutes in D + ( E ( J ) , C ) . The upper horizontal map is the natural specializationmap, and the vertical maps are the natural comparison isomorphisms in D + ( E ( J ) , C ) which are part of the mixed Hodge complexes Hdg • ( E ( J )) , resp. v ∗ J ψ Hf .Proof. In the global case I = J , Peters and Steenbrink defined a limit integralmixed Hodge complex of sheaves ψ Hf on E (see [23, 11.2.7]); we denote by ψ Hf ( C )its complex component. The complex ψ Hf ( C ) is defined as the simple complexassociated to the double complex A p,q = Ω p + q +1 (log E ) /W p Ω p + q +1 (log E ) where the differentials are defined in the same way as for A •• L . By [23, 11.3.1], wehave the following commutative diagram in D + ( E, C ) : C E spec −−−−→ Rψ f ( C ) y y Hdg • ( E ) C β −−−−→ ψ Hf ( C )The vertical maps are the natural comparison isomorphisms, and β is constructedby means of the residue maps: M | L | = p +1 ( a L ) ∗ Ω qE L ∼ = Gr Wp +1 Ω p + q +1 (log E ) → A p,q = Ω p + q +1 (log E ) /W p (Ω p + q +1 (log E ))Applying the functor a ∗ L : D + ( E, C ) → D + ( E L , C ), for any non-empty subset L of J , we get a commutative diagram in D + ( E L , C ) : C E L spec −−−−→ a ∗ L Rψ f ( C ) y y a ∗ L ( Hdg • ( E ) C ) β L −−−−→ a ∗ L ψ Hf ( C )We denote by b L the closed immersion E L → E ( J ), and we define an isomor-phism γ L : a ∗ L ( Hdg • ( E ) C ) → Ω • E L in D + ( E L , C ) as the composition of the naturalisomorphisms a ∗ L ( Hdg • ( E ) C ) ∼ = a ∗ L C E ∼ = C E L ∼ = Ω • E L in D + ( E L , C ).In view of Example 5.2, it suffices to prove the following claim: there exists acommutative diagram a ∗ L ( Hdg • ( E ) C ) β L −−−−→ a ∗ L ψ Hf ( C ) γ L y y δ L Ω • E L σ L −−−−→ a ∗ L C • L in D + ( E L , C ) , where the vertical arrows are isomorphisms, σ L is the map in (5.1),and δ L is a morphism in C + ( E L , C ) such that the composition v ∗ J Rψ f ( C ) ∼ = s (cid:3) ((( b L ) ∗ a ∗ L ψ Hf ( C )) L ∈ (cid:3) J ) s (cid:3) ( δ L ) −−−−→ s (cid:3) (( C • L ) L ∈ (cid:3) J ) = v ∗ J ψ Hf ( C ) coincides with the natural comparison isomorphism v ∗ J Rψ f ( C ) ∼ = v ∗ J ψ Hf ( C ) in D + ( E ( J ) , C ) . Consider the natural map δ ′ L : a ∗ L (Ω • (log E ) /W Ω • (log E )) → a ∗ L (Ω • (log E ) | E J /W Ω • (log E ) | E J )in C + ( E L , C ). Applying the functor a ∗ L to the diagram in Lemma 5.4, we seeimmediately that δ ′ L is a filtered quasi-isomorphism w.r.t. the quotient of the weightfiltrations. The map δ ′ L defines a morphism of double complexes a ∗ L A p,q → a ∗ L A p,qL ,inducing a map δ L on the associated simple complexes. Since δ ′ L is a filtered quasi-isomorphism, δ L is a filtered quasi-isomorphism w.r.t. the second filtration on thedouble complexes. The fact that σ L ◦ γ L = δ L ◦ β L follows from Lemma 5.4. (cid:3) INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 23
Proposition 5.6.
The natural specialization map Z → Rψ f ( Z ) induces canonicalisomorphisms Gr F H i ( E ( J ) , C ) ∼ = Gr F H i ( E ( J ) , v ∗ J Rψ f ( C )) for i ≥ . In particular, by restricting to the weight zero part, we get canonicalisomorphisms W H i ( E ( J ) , Q ) ∼ = W H i ( E ( J ) , v ∗ J Rψ f ( Q )) Proof.
It suffices to prove the statement after replacing rational coefficients bycomplex coefficients. Then the second isomorphism follows from the first sinceboth sides have Hodge type (0 , Gr F to the morphism σ L in (5.1), we get a morphism ofcomplexes Gr F ( σ L ) : ( a L ) ∗ O E L → Ω • +1 X (log E ) | E L /W • (Ω • +1 X (log E ) | E L ) = Gr W • +1 (Ω • +1 X (log E ) | E L )and this map is a quasi-isomorphism: by Lemma 5.4, the target is isomorphicto the complex ⊕ M ⊂ I, | M | = • +1 ( a L ∪ M ) ∗ O E L ∪ M . Passing to the associated simplecomplexes, we see that Gr F ( σ ) : Gr F Hdg • ( E ( J )) C → Gr F ψ Hf ( C )is a quasi-isomorphism, as well. (cid:3) Mixed Hodge structure on the nearby and vanishing cohomology.
Consider a complex variety X and a flat morphism f : X → Spec C [ t ] with smoothgeneric fiber. Let x be any complex point of X s . The cohomology spaces R i ψ f ( Z ) x carry a natural mixed Hodge structure [17, 15.13][23, § Y, g, ϕ ) of ( f, x ) (Definition 4.7), with Y s = P i ∈ I E i ; then ϕ − ( x ) = E ( J ) for some non-empty subset J of I . There are canonical isomorphisms H i ( E ( J ) , v ∗ J Rψ g ( Z )) ∼ = R i ψ f ( Z ) x for i ≥
0, and in this way R i ψ f ( Z ) x inherits an integral mixed Hodge structure,which does not depend on the chosen strictly semi-stable model.If we denote by R Θ f the vanishing cycles functor, then R i Θ f ( Z ) x also carries anatural mixed Hodge structure [18, 1.1]. We have R i Θ f ( Z ) x ∼ = R i ψ f ( Z ) x for i >
0, so R i Θ f ( Z ) x inherits a mixed Hodge structure. For i = 0, R i Θ f ( Z ) x carries a pure Hodge structure of weight zero.If X is smooth at x , then R i ψ f ( Z ) x is isomorphic to the degree i integral singularcohomology of the topological Milnor fiber of f at x for each i , so this cohomologycarries a mixed Hodge structure. If x is an isolated singularity of f , then this mixedHodge structure was constructed in [25]. Likewise, R i Θ f ( Z ) x is isomorphic to thedegree i reduced integral singular cohomology of the topological Milnor fiber of f at x .5.5. Non-archimedean interpretation of the weight zero subspace.
In thissection, we put R = C [[ t ]] and K = C (( t )), and we endow K with the absolutevalue | · | r for some fixed r ∈ ]0 , Theorem 5.7.
Let X be a complex variety, endowed with a flat morphism f : X → Spec C [ t ] with smooth generic fiber. Let x be a point of X ( C ) with f ( x ) = 0 .Denote by F x the analytic Milnor fiber of f at x . For each i ≥ , there exists canonical isomorphisms α : H i ( F x , Q ) → W R i ψ f ( Q ) x α ′ : e H i ( F x , Q ) → W R i Θ f ( Q ) x Proof.
Take a strictly semi-stable model (
Y, g, ϕ ) of ( f, x ) (Definition 4.7), denoteby d its ramification index, and by Y the t -adic completion of Y . Put E = ϕ − ( x ),and denote by v : E → Y s the closed immersion. Then ϕ induces an isomorphismof \ ( K r ) a -analytic spaces] E [ := sp − Y ( r ) ( E ) b × K r ( d ) \ ( K r ) a ∼ = F x and an isomorphism of mixed Hodge structures R i ψ f ( Q ) x ∼ = H i ( E, v ∗ Rψ g ( Q ))On the other hand, Berkovich proved in [8, Thm 1.1(c)] that there exists a canonicalisomorphism H i ( E an , Q ) ∼ = W H i ( E, Q ), and it follows from Proposition 4.4 andLemma 4.5 that there exists a canonical isomorphism H i ( E an , Q ) ∼ = H i ( ] E [ , Q ).Hence, we obtain an isomorphism H i ( F x , Q ) ∼ = W H i ( E, Q ) and, by Proposition5.6, an isomorphism H i ( F x , Q ) ∼ = W R i ψ f ( Q ) x A standard argument shows that this isomorphism does not depend on the chosensemi-stable model.The mixed Hodge structure on R ψ f ( Q ) x is pure of weight zero, so for i = 0 themap α is an isomorphism α : H ( F x , Q ) → R ψ f ( Q ) x Passing to reduced cohomology yields the natural isomorphism α ′ . (cid:3) Remark.
Using the same methods as in [3] one can generalize Theorem 5.7 asfollows: if f : X → Spec C [ t ] is a flat morphism of complex varieties, Z a propersubvariety of the special fiber, and Z the formal completion of f along Z , then thereis for each i ≥ α : H i ( Z ( r ) η , Q ) → W H i ( Z, Rψ f ( Q ))One reduces to the case where f has smooth generic fiber by taking a hypercoveringof f ; then one uses the proof of Theorem 5.7. (cid:3) Corollary 5.8. If X is smooth, of pure dimension n + 1 , and if we denote by s the dimension of the singular locus of f at x , then H i ( F x , Q ) = 0 for i / ∈ { , n − s, n − s + 1 , . . . , n } . If s < n , then F x is arc-connected.Proof. It is well-known that the reduced cohomology of the topological Milnor fiberof f at x vanishes for i / ∈ { n − s, n − s + 1 , . . . , n } . If s < n , then R ψ f ( Q ) x ∼ = Q ,so F x is connected and hence arc-connected by [4, 3.2.1]. (cid:3) Corollary 5.9.
We assume that X is smooth of pure dimension n + 1 with n > , f is projective and x is the only singular point of the special fiber X s = f − (0) . We INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 25 denote by X the t -adic completion of f . Then we have an isomorphism of longexact sequences −→ H i (( X s ) an , Q ) −→ H i ( X ( r ) η , Q ) −→ e H i ( F x , Q ) −→ H i +1 (( X s ) an , Q ) −→ γ y ∼ = β y ∼ = ∼ = y α ′ y ∼ = −→ W H i ( X s , Q ) −→ W H i ( X s , Rψ f ( Q )) −→ W R i Θ f ( Q ) x −→ W H i +1 ( X s , Q ) −→ where the upper long exact sequence is the one from Proposition 4.6, the lower oneis obtained by applying the exact functor Gr W to the canonical long exact sequenceof mixed Hodge structures [18, 1.1] . . . → H i ( X s , Q ) → H i ( X s , Rψ f ( Q )) → H i ( X s , R Θ f ( Q )) ∼ = R i Θ f ( Q ) x → . . . and where β is the isomorphism of [3, 5.1] and γ is the isomorphism of [8, Thm 1.1(c)] .This diagram breaks up in commutative squares of isomorphisms, for each ≤ i < n , H i (( X s ) an , Q ) ∼ −−−−→ H i ( X ( r ) η , Q ) γ y ∼ = ∼ = y β W H i ( X s , Q ) ∼ −−−−→ W H i ( X s , Rψ f ( Q )) as well as an isomorphism of exact sequences −→ H n (( X s ) an , Q ) −→ H n ( X ( r ) η , Q ) −→ H n ( F x , Q ) −→ H n +1 (( X s ) an , Q ) −→ γ y ∼ = β y ∼ = ∼ = y α y ∼ = −→ W H n ( X s , Q ) −→ W H n ( X s , Rψ f ( Q )) −→ W R n ψ f ( Q ) x −→ W H n +1 ( X s , Q ) −→ f as in Corollary 5.9; see [10, § Comparison with the motivic Milnor fiber
In this final section, we consider the motivic counterparts of the above results.Let k be a field of characteristic zero, and put R = k [[ t ]] and K = k (( t )). We fixan element r in ]0 ,
1[ and we endow K with the t -adic absolute value | · | r . Since r will remain fixed throughout this section, we simplify notation by writing K for thenon-archimedean field K r and X η for the generic fiber X ( r ) η of a special formal R -scheme in the category of K r -analytic spaces. For each integer d >
0, we put K ( d ) = K [ t d ] / (( t d ) d − t ) and we denote by R ( d ) the normalization of R in K ( d ).6.1. Motivic volume of a formal scheme.
Let X be a generically smoothspecial formal R -scheme. In [19, 7.39] we defined the motivic volume S ( X ; c K s ) of X . It is an element of the localized Grothendieck ring of X -varieties M X (seefor instance [21, § M X ). The aim of the current sectionis to explain the behaviour of the motivic volume under extension of the base ring R . This result will be used further on to give an expression of the motivic volumein terms of a semi-stable model of X (Theorem 6.11).As we noted in [19, 7.40], the motivic volume depends in general on the choiceof the uniformizer t , or, more precisely, on the K -fields K ( d ). If k is algebraicallyclosed, then up to K -isomorphism, K ( d ) is the unique extension of K of degree d ,and the motivic volume is independent of the choice of uniformizer. In order to speak about the motivic volume of a special formal R ( d )-schemewhen k is not algebraically closed, we fix the uniformizer t d in R ( d ). This yields anisomorphism of k -algebras R ( d ) ∼ = k [[ t d ]] and natural isomorphisms of R -algebras( R ( d ))( e ) ∼ = R ( d · e ) for d, e > L the class of the affine line A X in M X . This is a unitin M X . We’ll denote by R X the subring R X := M X (cid:20) T b T b − L a (cid:21) ( a,b ) ∈ Z × N ∗ of M X [[ T ]], and by R ′ X the subring of M X [[ T ]] consisting of elements of theform P ( T ) /Q ( T ), with P ( T ) , Q ( T ) polynomials over M X such that Q (0) is a unitin M X , Q ( T ) is monic, and the degree of Q ( T ) is at least the degree of P ( T ). Thering R ′ X contains R X . There exists a unique morphism of M X -algebraslim T →∞ : R ′ X → M X mapping P ( T ) /Q ( T ) (with P ( T ) , Q ( T ) as above) to the coefficient of T deg Q in P ( T ), where deg Q denotes the degree of Q ( T ). It restricts to the morphismlim T →∞ : R X → M X from [19, 7.35].For any ring A , any element a ( T ) = P i ≥ a i T i of A [[ T ]], and any integer d > a ( T )[ d ] = P i ≥ , d | i a i T i ∈ A [[ T ]]. Lemma 6.1.
For any ( p, q, r ) ∈ Z × N × N ∗ with q ≤ r , T q T r − L p ∈ R X Proof.
For any ( p, q, r ) ∈ Z × N × N ∗ we use the notation D pq,r := T q T r − L p ∈ M X [[ T ]]By definition, D pr,r ∈ R X . Moreover, the relation D p ,r = L − p (cid:0) D pr,r − (cid:1) showsthat D p ,r ∈ R X .Put I = { ( q, r ) ∈ N × N ∗ | q ≤ r } . We will show that D pq,r ∈ R X for any( p, q, r ) ∈ Z × I , by induction on ( q, r ) w.r.t. the lexicographic ordering on I . Wemay assume that 0 < q < r . We have D pq,r = T r T r − L p · T r − q − − T r − L p · T q T r − q − ≡ − D p ,r · D q,r − q mod R X We know that D p ,r ∈ R X , and if q ≤ r − q , then D q,r − q ∈ R X by the inductionhypothesis. Hence, we may assume that q > r − q . Then we write D p ,r · D q,r − q = D p q − r,r · D r − q,r − q and since 0 < q − r < q the induction hypothesis implies that D p q − r,r belongs to R X . (cid:3) INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 27
Lemma 6.2.
For any integer d > , the morphism of M X -modules φ d : M X [[ T ]] → M X [[ T ]] : a ( T ) a ( T )[ d ] restricts to a M X -module endomorphism of R X . Moreover, lim T →∞ ◦ φ d = lim T →∞ Proof.
Consider an integer n >
0, a tuple a ∈ Z n , and a tuple b ∈ ( N ∗ ) n . We put C a,b := n Y i =1 T b i T b i − L a i ! [ d ] ∈ M X [[ T ]]It suffices to show that C a,b ∈ R X and thatlim T →∞ C a,b = 1For any pair of tuples u, v ∈ Z n , we put u · v = P ni =1 u i · v i . For each i ∈ { , . . . , n } ,we put m i = lcm ( b i , d ) and e i = m i /b i . If we put S = { u ∈ N n | ≤ u i ≤ e i for all i, and d | u · b } then the map S × N n → { w ∈ ( N ∗ ) n | w · b ∈ d N } : ( u, v ) ( u + v e , . . . , u n + v n e n )is a bijection. Therefore C a,b = X u ∈ S L − a · u T b · u ! · n Y i =1 L a i e i T b i e i − L a i e i ! Since e = ( e , . . . , e n ) belongs to S , the first factor is a polynomial with leadingterm L − a · e T b · e , so we see that C a,b belongs to R ′ X andlim T →∞ C a,b = 1as required. Lemma 6.1 shows that C a,b ∈ R X . (cid:3) Proposition 6.3.
For any generically smooth special formal R -scheme X and anyinteger d > , S ( X ; c K s ) = S ( X × R R ( d ); \ K ( d ) s ) in M X .Proof. By [19, 7.38+39] we may assume that X η admits a X -bounded gauge form ω (in the sense of [19, 2.11]). For each n > ω ( n ) the pull-back of ω to X η × K K ( n ). This is a X × R R ( n )-bounded gauge form.In [19, 4.9] we defined the volume Poincar´e series S ( X , ω ; T ) of ( X , ω ) by S ( X , ω ; T ) = X n> Z X × R R ( n ) | ω ( n ) | ! T n ∈ M X [[ T ]](the coefficients are motivic integrals). It is clear from the definition (and from ourchoice of uniformizer in R ( d )) that S ( X × R R ( d ) , ω ( d ); T ) = S ( X , ω ; T )[ d ]We showed in [19, 7.14] that S ( X , ω ; T ) is rational; more precisely, it belongs to R X . By definition, S ( X ; c K s ) = − lim T →∞ S ( X , ω ; T ) By Lemma 6.2, S ( X ; c K s ) = − lim T →∞ S ( X , ω ; T )= − lim T →∞ S ( X × R R ( d ) , ω ( d ); T )= S ( X × R R ( d ); \ K ( d ) s ) (cid:3) Proposition 6.4.
Let k ′ be a field containing k , and put R ′ = k ′ [[ t ]] . Let X be a generically smooth special formal R -scheme, and put X ′ = X b × R R ′ . Then S ( X ′ ; c K s ) is the image of S ( X ; c K s ) under the base change morphism M X →M X ′ .Proof. This is obvious from the definition of the motivic volume. (cid:3)
Motivic volume of a rigid variety.
Let X be a generically smooth specialformal R -scheme, and assume that X is stf t or that X η admits a universallybounded gauge form in the sense of [19, 7.41]. Such a universally bounded gaugeform exists, in particular, if X is the formal spectrum of a regular local R -algebra[19, 7.23+42].In [21, 8.3] and [19, 7.43] we defined the motivic volume S ( X η ; c K s ) of X η asthe image of S ( X ; c K s ) under the forgetful morphism M X → M k and we showedthat this definition only depends on X η and not on the model X . Proposition 6.5.
Let d > be an integer and X a separated smooth rigid K -variety. Assume that X and X × K K ( d ) admit universally bounded gauge forms,or that X is quasi-compact. Then S ( X × K K ( d ); \ K ( d ) s ) = S ( X ; c K s ) in M k .Proof. This follows immediately from Proposition 6.3. (cid:3)
Proposition 6.6.
Let k ′ be any field containing k , and put L = k ′ (( t )) . Let X bea separated smooth rigid K -variety. Assume that X and X × K L admit universallybounded gauge forms, or that X is quasi-compact. Then S ( X × K L ; c L s ) is the imageof S ( X ; c K s ) under the base change morphism M k → M k ′ .Proof. This follows immediately from Proposition 6.4. (cid:3)
Remark.
Even though the valuation on c K a is not discrete and the construction ofthe motivic volume does not apply to c K a -analytic spaces, the above results justifythe hope that one can associate a motivic volume V ol ( X ) to separated smoothquasi-compact rigid varieties X over c K a (or an even larger class of c K a -analyticspaces). This volume should have the property that V ol ( Y b × K c K a ) = S ( Y ; c K s )when Y is a separated smooth quasi-compact rigid K -variety, and opens the wayto a theory of motivic integration on c K a -analytic spaces. (cid:3) INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 29
Motivic Milnor fiber.
We start with an auxiliary result.
Lemma 6.7.
For any morphism of generically smooth special formal R -schemes h : Y → X such that h η is an isomorphism, we have S ( Y ; c K s ) = S ( X ; c K s ) in M X . Here the left hand side is viewed as an element of M X via the forgetfulmorphism M Y → M X induced by h .Proof. This is straightforward from the definition of the motivic volume in [19,7.39]. (cid:3)
Let X be a k -variety of pure dimension m , and let f : X → A k = Spec k [ t ] be aflat morphism with smooth generic fiber. We fix a closed point x on the special fiber X s of f . We denote by X the t -adic completion of f , and by X x the completionof f at x , i.e. the special formal R -scheme Spf b O X,x . Then ( X x ) η = F x is theanalytic Milnor fiber of f at x . We view F x as a k ′ (( t ))-analytic space, with k ′ theresidue field of x .If X is smooth at x then X x is the formal spectrum of a regular local R -algebra,so S ( F x ; c K s ) is defined and equal to S ( X x ; c K s ) ∈ M x . If X is not smooth at x , itis not clear if F x admits a universally bounded gauge form. However, we still havethe following property. Proposition 6.8.
The analytic Milnor fiber F x , viewed as a k ′ (( t )) -analytic space,uniquely determines the motivic volume S ( X x ; c K s ) ∈ M x Proof.
The normalization morphism f X x → X x is a morphism of special formal R -schemes which induces an isomorphism between the generic fibers because F x isnormal (even smooth) and normalization commutes with taking generic fibers [11,2.1.3]. By Lemma 6.7, S ( f X x ; c K s ) = S ( X x ; c K s ) ∈ M x Moreover, the k ′ [[ t ]]-algebra of power-bounded analytic functions on F x is canoni-cally isomorphic to O ( f X x ) by [12, 7.4.1], so F x determines S ( X x ; c K s ). (cid:3) Definition 6.9. If Y is a generically smooth special formal R -scheme of purerelative dimension d , then we define the motivic nearby cycles S Y of Y by S Y = L d · S ( Y ; c K s ) ∈ M Y Let X be a k -variety of pure dimension, f : X → Spec k [ t ] a flat morphism withsmooth generic fiber, and x a closed point of the special fiber X s of f . Denote by X /R the t -adic completion of f and by X x /R the formal completion of f at x .We define the motivic Milnor fiber S f,x of f at x by S f,x = S X x ∈ M x and the motivic nearby cycles S f of f by S f = S X ∈ M X If X is smooth, then Denef and Loeser defined the motivic nearby cycles of f and the motivic Milnor fiber of f at x in [14, § k = C and X is smooth, S f and S f,x have the same Hodge realization as thecohomological nearby cycles, resp. Milnor fiber, in an appropriate Grothendieckring of mixed Hodge modules with monodromy action; see [13, 4.2.1] and [14,3.5.5].6.4. Expression in terms of a semi-stable model.
Now we come to the mainresult of this section: an expression for the motivic volume of a formal scheme interms of a semi-stable model. We’ll freely use the notation and terminology from[19, § Definition 6.10.
Let U be a generically smooth special formal R -scheme. Astrictly semi-stable model ( V , g ) for U consists of the following data: (1) an integer d > , which we call the ramification index of the model ( V , g ) , (2) a regular special formal R ( d ) -scheme V , such that V s is a reduced strictnormal crossing divisor on V , (3) a morphism of special formal R -schemes g : V → U which induces anisomorphism of K ( d ) -analytic spaces V η ∼ = U η × K K ( d ) . Such a strictly semi-stable model exists, in particular, when Y is the formalcompletion of a generically smooth stf t formal R -scheme along a closed subschemeof its special fiber (see the remark following Theorem 4.10). Theorem 6.11. If U is a generically smooth special formal R -scheme and ( V , g ) is a strictly semi-stable model for U , with V s = P i ∈ I E i , then (6.1) S U = X ∅6 = J ⊂ I (1 − L ) | J |− [ E oJ ] ∈ M U In particular, the right hand side does not depend on the chosen strictly semi-stablemodel.Proof.
By Proposition 6.3, we may assume that the ramification index d of thestrictly semi-stable model is equal to one, i.e. that g η : V η → U η is an isomorphism.Then by Lemma 6.7 we have S U = S V in M U , and it follows from [19, 7.36] that S V = X ∅6 = J ⊂ I (1 − L ) | J |− [ E oJ ] ∈ M V (cid:3) Remark.
If the formal R -scheme V in the statement of Theorem 6.11 is stf t ,then E oJ coincides with the stratum ( V s ) oJ of the strictly semi-stable k -scheme V s (Section 2.2) for any ∅ 6 = J ⊂ I = Irr ( V s ). (cid:3) Corollary 6.12.
Consider f : X → Spec k [ t ] , x and X x as in Definition 6.9. If ( V , g ) is a strictly semi-stable model for X x , with V s = P i ∈ I E i , then (6.2) S f,x = X ∅6 = J ⊂ I (1 − L ) | J |− [ E oJ ] ∈ M x In particular, the right hand side does not depend on the chosen strictly semi-stablemodel.
INGULAR COHOMOLOGY OF THE ANALYTIC MILNOR FIBER 31
Note in particular that any strictly semi-stable model (
Y, g, ϕ ) of the germ ( f, x )(in the sense of Definition 4.7) induces a strictly semi-stable model for X x by takingthe formal completion of Y along ϕ − ( x ) (of course, the projectivity condition canbe omitted in Definition 4.7). Therefore, (6.2) also gives an expression for S f,x interms of a semi-stable model of ( f, x ).As a special case, if X is smooth, we get an expression for Denef and Loeser’smotivic nearby cycles and motivic Milnor fiber in terms of a strictly semi-stablemodel of f , resp. of ( f, x ). This expression is not clear from their definition [14, § U , resp. X x . Remark.
Theorem 6.11 makes it possible to compare our notion of motivic nearbycycles and of motivic volume of a rigid variety with the ones introduced by Ayoub[1][2], after specialization to an appropriate Grothendieck ring of k -motives. Detailswill appear elsewhere. (cid:3) Theorem 6.11 is similar in spirit to Theorem 4.10 and the subsequent remark.However, since motivic integrals take their values in a Grothendieck ring, all “non-additive” information is lost when taking the motivic volume S ( X x ; c K s ). As we’veseen, non-archimedean geometry provides a powerful additional tool to prove inde-pendence results of this type. References [1] J. Ayoub.
Les motifs des vari´et´es rigides analytiques . preprint, 2008.[2] J. Ayoub.
Les six op´erations de Grothendieck et le formalisme des cycles ´evanescents dansle monde motivique . to appear, 2008.[3] V. G. Berkovich. A non-Archimedean interpretation of the weight zero subspaces of limitmixed Hodge structures. In
Algebra, Arithmetic and Geometry - Manin Festschrift (to ap-pear) . Boston: Birkh¨auser.[4] V. G. Berkovich.
Spectral theory and analytic geometry over non-archimedean fields , vol-ume 33 of
Mathematical Surveys and Monographs . AMS, 1990.[5] V. G. Berkovich. ´Etale cohomology for non-Archimedean analytic spaces.
Publ. Math., Inst.Hautes ´Etud. Sci. , 78:5–171, 1993.[6] V. G. Berkovich. Vanishing cycles for formal schemes II.
Invent. Math. , 125(2):367–390, 1996.[7] V. G. Berkovich. Smooth p -adic analytic spaces are locally contractible. Invent. Math. ,137(1):1–84, 1999.[8] V. G. Berkovich. An analog of Tate’s conjecture over local and finitely generated fields.
Int.Math. Res. Not. , 2000(13):665–680, 2000.[9] P. Berthelot. Cohomologie rigide et cohomologie rigide `a supports propres.
Prepublication,Inst. Math. de Rennes , 1996.[10] E. Brieskorn. Die Monodromie der isolierten Singularit¨aten von Hyperfl¨achen.
Manuscr.Math. , 2:103–161, 1970.[11] B. Conrad. Irreducible components of rigid spaces.
Ann. Inst. Fourier , 49(2):473–541, 1999.[12] A. J. de Jong. Crystalline Dieudonn´e module theory via formal and rigid geometry.
Publ.Math., Inst. Hautes ´Etud. Sci. , 82:5–96, 1995.[13] J. Denef and F. Loeser. Motivic Igusa zeta functions.
J. Algebraic Geom. , 7:505–537, 1998,arxiv:math.AG/9803040.[14] J. Denef and F. Loeser. Geometry on arc spaces of algebraic varieties.
Progr. Math. , 201:327–348, 2001, arxiv:math.AG/0006050.[15] P. Gabriel and M. Zisman.
Calculus of fractions and homotopy theory , volume 35 of
Ergeb-nisse der Math. und ihrer Grenzgebiete . Springer-Verlag, 1967. [16] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat.
Toroidal embeddings 1 , volume339 of
Lecture Notes in Mathematics . Springer-Verlag, 1973.[17] V. Navarro Aznar. Sur la th´eorie de Hodge-Deligne.
Invent. Math. , 90:11–76, 1987.[18] V. Navarro Aznar. Sur les structures de Hodge mixtes associ´ees aux cycles evanescents. In
Hodge theory, Proc. U.S.-Spain Workshop, Sant Cugat/Spain 1985 , volume 1246 of
Lect.Notes Math. , pages 143–153, 1987.[19] J. Nicaise. A trace formula for rigid varieties, and motivic Weil generating series for formalschemes. to appear in Math. Ann. [20] J. Nicaise and J. Sebag. Invariant de Serre et fibre de Milnor analytique.
C.R.Ac.Sci. ,341(1):21–24, 2005.[21] J. Nicaise and J. Sebag. The motivic Serre invariant, ramification, and the analytic Milnorfiber.
Invent. Math. , 168(1):133–173, 2007.[22] J. Nicaise and J. Sebag. Rigid geometry and the monodromy conjecture. In D. Ch´eniotet al., editor,
Singularity Theory, Proceedings of the 2005 Marseille Singularity School andConference , pages 819–836. World Scientific, 2007.[23] C. Peters and J.H.M. Steenbrink.
Mixed Hodge structures , volume 52 of
Ergebnisse der Math-ematik und ihrer Grenzgebiete. 3. Folge . Berlin: Springer, 2008.[24] J.H.M. Steenbrink. Limits of Hodge structures.
Invent. Math. , 31:229–257, 1976.[25] J.H.M. Steenbrink. Mixed Hodge structure on the vanishing cohomology. In P. Holm, editor,
Real and complex Singularities, Proc. Nordic Summer Sch., Symp. Math., Oslo 1976 , pages525–563. Alphen a.d. Rijn: Sijthoff & Noordhoff, 1977.[26] M. Temkin. Desingularization of quasi-excellent schemes in characteristic zero. to appear inAdv. Math. , arXiv:math/0703678.[27] A. Thuillier. G´eom´etrie toro¨ıdale et g´eom´etrie analytique non archim´edienne. Application autype d’homotopie de certains sch´emas formels.
Manuscr. Math. , 123(4):381–451, 2007.
Universit´e Lille 1, Laboratoire Painlev´e, CNRS - UMR 8524, Cit´e Scientifique, 59655Villeneuve d’Ascq C´edex, France
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