Motivic and analytic nearby fibers at infinity and bifurcation sets
aa r X i v : . [ m a t h . AG ] O c t MOTIVIC AND ANALYTIC NEARBY FIBERS AT INFINITY ANDBIFURCATION SETS by Lorenzo Fantini & Michel Raibaut
Abstract . —
In this paper we use motivic integration and non-archimedean analytic geometry tostudy the singularities at infinity of the fibers of a polynomial map f : A d C → A C . We show that themotive S ∞ f,a of the motivic nearby cycles at infinity of f for a value a is a motivic generalization ofthe classical invariant λ f ( a ), an integer that measures a lack of equisingularity at infinity in the fiber f − ( a ). We then introduce a non-archimedean analytic nearby fiber at infinity F ∞ f,a whose motivicvolume recovers the motive S ∞ f,a . With each of S ∞ f,a and F ∞ f,a can be naturally associated a bifurcationset; we show that the first one always contains the second one, and that both contain the classicaltopological bifurcation set of f if f has isolated singularities at infinity. October 16, 2018
Contents
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Motivic integration and nearby cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. Motivic nearby cycles at infinity and the motivic bifurcation set. . . . . . . . . . . . . . 74. Analytic nearby fiber at infinity and the Serre bifurcation set. . . . . . . . . . . . . . . . . 11References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1. Introduction
Let U be a smooth and irreducible complex algebraic variety and let f : U → A C be a dominantmap. It is well known that there exists a finite subset B of A C ( C ) such that the map f | U \ f − ( B ) : U \ f − ( B ) −→ A C \ B is a locally trivial C ∞ -fibration, see for example [ ]. The smallest such subset B is called the topological bifurcation set of f and is denoted by B top f . The topological bifurcation set of f containsthe discriminant disc( f ) of f , that is the set f (cid:0) crit( f ) (cid:1) ⊂ A C ( C ) image of its set of critical points,but the inclusion might be strict since f needs not to be proper. For example, the map f : A C → A C defined by the polynomial f ( x, y ) = x ( xy −
1) is smooth, but has bifurcation set { } ; its fiber over0 is the only one to be disconnected. This is related to the fact that, loosely speaking, f ( x, y ) mayhave some “critical points at infinity”; one way of seeing this is to observe that the compactification LORENZO FANTINI & MICHEL RAIBAUT of the fiber f − ( a ) in P C will have a multibranch singularity for a = 0, and a cuspidal singularityotherwise.Assume that f : A d C → A C is a polynomial map. In some special cases, knowing the (compactlysupported) Euler characteristic of a fiber f − ( a ) is sufficient to determine whether a belongs to B top f . Indeed, if we denote by f − ( a gen ) a general fiber of f , we have(1.1) B top f = (cid:8) a ∈ A C ( C ) (cid:12)(cid:12) χ c (cid:0) f − ( a ) (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1)(cid:9) in the case of curves, that is whenever d = 2, see [ ], and, more generally, whenever f has isolatedsingularities at infinity (that is, the closure of a general fiber f − ( a gen ) of f in P d C has isolatedsingularities), see [ ].If the polynomial f ∈ C [ x , . . . , x d ] has isolated singularities in A d C (which does not necessarilymean that it has also isolated singularities at infinity), then Artal Bartolo–Luengo–Melle-Hern´andezproved in [ ] that the Euler characteristic of each fiber f − ( a ) can be computed in terms of theEuler characteristic of f − ( a gen ) and of some local numerical invariants, as we will now explain.To account for the singularities of the fibers, let µ a ( f ) be the sum of the Milnor numbers of thesingular points of f − ( a ), so that a belongs to the discriminant of f if and only if µ a ( f ) >
0, andset µ ( f ) = P a ∈ A C µ a ( f ). On the other hand, to keep track of the behavior of f at infinity, consider λ a ( f ) = µ (cid:0) P n , f − ( a ) , D (cid:1) − µ (cid:0) P d , f − ( a gen ) , D (cid:1) , where f − ( a ) is the closure f − ( a ) in P d , D isthe divisor cut out by the homogeneous part of f of degree deg( f ) in the hyperplane at infinity of P d , and µ (cid:0) P n , f − ( b ) , D (cid:1) is the generalized Milnor number of Parusi´nski from [ ]. The invariants λ a ( f ) vanish for all but finitely many values of a , therefore we can set λ ( f ) = P a ∈ A C λ a ( f ). By [ ,Theorem 1.7], we then have the following equalities: χ c (cid:0) f − ( a ) (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1) + ( − d (cid:0) µ a ( f ) + λ a ( f ) (cid:1) , (1.2) χ c (cid:0) f − ( a gen ) (cid:1) = 1 + ( − d − (cid:0) µ ( f ) + λ ( f ) (cid:1) . (1.3)Similar results were obtained in [ ] and [ ], and in the case of curves in [ ], [ ], and [ ]. Inparticular, whenever the condition 1.1 also holds one can deduce that B top f = (cid:8) a ∈ A C ( C ) (cid:12)(cid:12) λ a ( f ) = 0 (cid:9) ∪ disc( f ) . The invariant λ a ( f ) gives some rudimentary measure of the lack of equisingularity of f at infinityat the value a . Since it is defined using Euler characteristics, it is natural to expect to be able togeneralize λ a ( f ) to a motive in a Grothendieck ring of C -varieties by using Denef and Loeser’smotivic integration. In that direction, and with the goal of studying singularities at infinity, thesecond author defined in [ ] a motive S ∞ f,a , the so called motivic nearby cycles at infinity of f for the value a , living in a localization M ˆ µ C of the Grothendieck ring of C -varieties with a goodˆ µ -action. In order to do so, he used motivic integration techniques, following work of Denef–Loeser[ ], Bittner [ ], and Guibert–Loeser–Merle [ ]. This allowed him to define a motivic bifurcationset B mot f of f as the union of the discriminant of f with the set of values whose motivic nearbycycles at infinity do not vanish. This is a finite set, and it coincides with B top f if f is a convenientand non degenerate polynomial with respect to its Newton polygon at infinity (in the sense of [ ]),as is shown in [ , Theorem 4.8]. However, no explicit link between S ∞ f,a and λ a ( f ) was establishedin [ ].The first contribution of this paper is to prove that S ∞ f,a is indeed a motivic generalization of thenumerical invariant λ a ( f ), whenever the latter is defined. In particular, we obtain the followingresult: OTIVIC AND ANALYTIC NEARBY FIBERS AT INFINITY AND BIFURCATION SETS Theorem . — Let f be a polynomial in C [ x , . . . , x d ] with isolated singularities in A d C . For anyvalue a in A C ( C ), we have χ c (cid:0) S ∞ f,a (cid:1) = ( − d − λ a ( f ) . We obtain this result as a consequence of a more general equality between Euler characteristic ofmotives, proven in Theorem 3.3, which in particular does not require f to have isolated singularities.As an application, we deduce the following fact: Theorem . — Let f be a polynomial in C [ x , . . . , x d ]. Then we have (cid:8) a ∈ A C ( C ) (cid:12)(cid:12) χ c (cid:0) f − ( a ) (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1)(cid:9) ⊂ B mot f . In particular, B mot f contains B top f whenever the condition 1.1 is satisfied, for example if f hasisolated singularities at infinity.Let us briefly explain the definition of S ∞ f,a and give a short outline of the proof of the theorems.We can work in a slightly more general setting and assume that k is a field of characteristic zerocontaining all roots of unity, U is a smooth k -variety, and f : U → A k is a dominant morphism.We consider a compactification ( X, ˆ f ) of ( U, f ), that is a k -variety X containing U as a dense opensubset, together with a proper extension ˆ f of f to X . As is usual in motivic integration, the motive S ∞ f,a is defined as a limit of a generating series P mes( X i ) T i , where mes( X i ) is the motivic measureof a subset X i of the scheme of arcs L ( X ) of X . Namely, X i consists of arcs that have origin atinfinity, that is in X \ U , order of contact i along the fiber ˆ f − ( a ), and are only mildly tangentto X \ U . As such, S ∞ f,a can be expressed as the difference of the images in M ˆ µk of S ˆ f − a,U , themotivic nearby cycles of ˆ f − a supported by U , a motive defined by Guibert–Loeser–Merle [ ]using a motivic integration procedure as above but imposing no condition on the origin of arcs, bythe motivic nearby fiber S f − a of Denef–Loeser [ ], which is constructed from arcs with origin in U . We show in Theorem 3.3 that the resulting equality generalizes the formula 1.2, by taking theEuler characteristics and using realization results for S ˆ f − a,U and S f − a from [ ] and [ ]. The twotheorems we stated above follow then from simple computations.The second contribution of this paper is the construction of a non-archimedean analytic versionof S ∞ f,a . Namely, the analytic nearby fiber F ∞ f,a of f for the value a is the analytic space over thefield K = k (( t )) (endowed with a given t -absolute value) defined as F ∞ f,a = ( U K ) an \ (cid:0) c U R (cid:1) i , where (cid:0) c U R (cid:1) i is the space of analytic nearby cycles of [ ], that is the analytic space associated tothe formal completion of U along the fiber f − ( a ), and ( U K ) an is the analytification of the basechange of U to K via the map f − a . This definition is analogous to the one of the motive S ∞ f,a ,since if ( X, ˆ f ) is a compactification of ( U, f ) then F ∞ f,a consists of those points in the analytification( X K ) an of X K that have origin on X K \ U K but are not completely contained in X K \ U K . Oneadvantage of F ∞ f,a over S ∞ f,a is that, while the latter is defined in terms of an extension of f toa compactification of U , and only later proven to be independent of such a choice, the former isdefined exclusively in terms of f .We then construct a motivic specialization of F ∞ f,a , its Serre invariant S (cid:0) F ∞ f,a (cid:1) , and use it todefine a third bifurcation set, the Serre bifurcation set B ser f of f , as the union of the discriminantof f with the set of values a such that S (cid:0) F ∞ f,a (cid:1) does not vanish. LORENZO FANTINI & MICHEL RAIBAUT
Using the decomposition of F ∞ f,a as a difference, and realization results for the volume and Serreinvariant of analytic spaces from [
26, 5, 16 ], in the subsections 4.4 and 4.5 we prove the followingresults:
Theorem . — Let U be a smooth connected k -variety and let f : U → A k be a dominant morphism.For any value a in A k ( k ), we have:1. Vol (cid:0) F ∞ f,a (cid:1) = L − (dim U − S ∞ f,a in M ˆ µk ;2. S (cid:0) F ∞ f,a (cid:1) = S ∞ f,a mod ( L −
1) in M ˆ µk / ( L − (cid:8) a ∈ A C ( C ) (cid:12)(cid:12) χ c (cid:0) f − ( a ) (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1)(cid:9) ⊂ B ser f ⊂ B mot f . We expect that a deep study of the geometry (and ´etale cohomology) of the analytic nearbyfibers at infinity will lead to a better understanding of various phenomena of equisingularity atinfinity, and we plan to study this topic further in an upcoming project.
Acknowledgments. —
We are very thankful to David Bourqui, Raf Cluckers, Johannes Nicaise,and Julien Sebag, who organized the conference “Nash : Sch´emas des arcs et singularit´es” inRennes in 2016, gave the first of us the opportunity to give a talk there, and proposed us tocontribute to the conference proceedings. We are also grateful to Emmanuel Bultot, PierretteCassou-Nogu`es, Alexandru Dimca, Johannes Nicaise, and Claude Sabbah for inspiring discussions,and the anonymous referee for his comments and corrections. This work is partially supported byANR-15-CE40-0008 (D´efig´eo).
Notations. —
In the paper we will freely use the following notations. – k is a field of characteristic zero that contains all the roots of unity. – A k -variety is a separated k -scheme of finite type. – ˆ µ is the projective limit lim ← µ n , where for any positive integer n we denote by µ n the groupscheme of n -th roots of unity. – R is the discrete valuation ring k [[ t ]], endowed with the t-adic valuation. – K = k (( t )) is the fraction field of R , endowed with a fixed t -adic absolute value. – For any integer n , we denote by K ( n ) = K (( t /n )) the unique extension of degree n of K obtained by joining a n -th root of t to K , and by R ( n ) the normalization of R in K ( n ). – Given a k -variety X and a morphism f : X → A k = Spec k [ t ], we set X R = X × A k Spec R and X K = X × A k Spec K .
2. Motivic integration and nearby cycles
In this section we review some basic constructions in motivic integration that will be usedthroughout the paper. We refer to [ ], [ ], [ ], [ ], [ ], and [ ] for a more thorough discussionof these notions. We say that an action of µ n on a k -variety X is good if every µ n -orbit is contained in some open affine subscheme of X , and that an action of ˆ µ on X is good if itfactors through a good µ n -action for some n . If S is a k -variety, we denote by Var ˆ µS the category of S -varieties with a good ˆ µ -action, that is the category whose objects are the k -varieties X endowed OTIVIC AND ANALYTIC NEARBY FIBERS AT INFINITY AND BIFURCATION SETS with a good action of ˆ µ and with a morphism X → S that is equivariant with respect to the trivialˆ µ -action on S , and whose morphisms are ˆ µ -equivariant morphisms over S .We denote by K (Var ˆ µS ) the Grothendieck ring of Var ˆ µS . It is defined as the abelian groupgenerated by the isomorphism classes [ X, σ ] of the elements (
X, σ ) of Var ˆ µS , with the relations[ X, σ ] = [
Y, σ | Y ] + [ X \ Y, σ | X \ Y ] if Y is a closed subvariety of X stable under the action of ˆ µ , andmoreover [ X × A nk , σ ] = [ X × A nk , σ ′ ] if σ and σ ′ are two liftings of the same ˆ µ -action on X to anaffine action on X × A kn . There is a natural ring structure on K (Var ˆ µS ), the product being inducedby the fiber product over S . In the rest of the paper we will simply write [ X ] for [ X, σ X ], as norisk of confusion will arise.We denote by M ˆ µS the localization of K (Var ˆ µS ) at the element L S = [ A S ], that is the class of theaffine line over S endowed with the trivial action. If S = Spec k we simply write M ˆ µk and L for M ˆ µS and L S respectively, and if we only consider varieties with trivial ˆ µ -action, we obtain analogousGrothendieck rings M k and M S .If f : S ′ → S is a morphism of k -varieties, then the composition with f on the left induces a direct image group morphism f ! : M ˆ µS ′ → M ˆ µS , while taking the fiber product with S ′ over S induces an inverse image ring morphism f ∗ : M ˆ µS → M ˆ µS ′ . We denote by M ˆ µS [[ T ]] rat the M ˆ µS -submodule of M ˆ µS [[ T ]] generated by 1 and by the finite prod-ucts of terms p e,i ( T ) = L e T i / (1 − L e T i ), with e in Z and i in Z > . The formal series in M ˆ µS [[ T ]] rat are said to be rational . There exists a unique M ˆ µS -linear morphism lim T →∞ : M ˆ µS [[ T ]] rat → M ˆ µS such that for any finite subset ( e i , j i ) I ∈ i of Z × Z > we have lim T →∞ (cid:0) Q I p e i ,j i ( T ) (cid:1) = ( − | I | . Let X be a k -variety of dimension d . We denote by L n ( X ) the space of n -jets of X , that is the k -scheme of finite type whose functor of points is the following:for every k -algebra L , the L -points of L n ( X ) are the morphisms Spec (cid:0) L [ s ] / ( s n +1 ) (cid:1) → X . Inparticular, L ( X ) is canonically isomorphic to X itself. Right compositions with the canonicalprojections L [ s ] / ( s n +1 ) → L [ s ] / ( s n ) yield morphisms L n ( X ) → L n − ( X ), making {L n ( X ) } n intoa projective system. These morphisms are A dk -bundles when X is smooth of pure dimension d . The arc space of X , denoted by L ( X ), is the projective limit of the system {L n ( X ) } n ; we denote by π n : L ( X ) → L n ( X ) the canonical projections. The arc space of X is a k -scheme that is generallynot of finite type. For every finite extension k ′ of k the k ′ -rational points of L ( X ) parametrize themorphisms Spec k ′ [[ s ]] → X . If ϕ ∈ L ( X ) is an arc on X , the point π ( ϕ ) ∈ X is called the origin of ϕ on X . Observe that µ n acts canonically on L n ( X ) and on L ( X ) by setting λ.ϕ ( s ) = ϕ ( λs ).Assume that we have a morphism f : X → A k = Spec k [ t ]. With a L -arc ϕ : Spec L [[ s ]] → X on X we can then associate its order ord f ( ϕ ) := ord s (cid:0) ϕ ( f ) (cid:1) ∈ Z ≥ ∪ { + ∞} , and its angularcomponent ac f ( ϕ ) ∈ L , defined as the leading coefficient of the power series ϕ ( f ) ∈ L [[ s ]]; byconvention ac f ( ϕ ) = 0 if ϕ ( f ) = 0.More generally, if F is a closed subvariety of X of coherent ideal sheaf I F , then the contact order of an arc ϕ ∈ L ( X ) along F is ord F ( ϕ ) := inf g ord ϕ ( g ), where g runs among the local sections of I F at the origin of ϕ . It is greater than zero if and only if the origin of ϕ lies in F , while it takesthe value infinity if ϕ is an arc on F . For example, if we have a morphism f : X → A k as abovethen the contact order of ϕ along the fiber f − (0) is precisely ord f ( ϕ ). LORENZO FANTINI & MICHEL RAIBAUT
Let X be a purely dimensional k -variety, let f : X → A k be amorphism, and denote by X ( f ) the zero locus of f in X . By work of Denef–Loeser [
8, 11 ], Bittner[ ], and Guibert–Loeser–Merle [ ], there exists a group morphism S f : M X −→ M ˆ µX ( f ) called the motivic nearby cycles morphism of f . In particular, for every open immersion i : U → X we obtain a motive S f,U := S f ( i ) in M ˆ µX ( f ) , the motivic nearby cycles of f supported on U .We will recall the construction of the motive S f,U as done in [ ], restricting to the case where U is a smooth dense open subvariety of X , since this case will play an important role in the rest ofthe paper. We denote by F = X \ U the complement of U in X , and by passing to a resolution ofthe singularities of the pair ( X, F ) we can assume without loss of generality that X is itself smooth.For any two positive integers n and δ , we consider the subscheme X δn ( f, U ) := (cid:8) ϕ ∈ L ( X ) (cid:12)(cid:12) ord f ( ϕ ) = n, ac f ( ϕ ) = 1 , ord F ( ϕ ) ≤ nδ (cid:9) of L ( X ). Since ord f ( ϕ ) >
0, we have that π (cid:0) X δn ( f, U ) (cid:1) ⊂ X ( f ), and so for any m ≥ n the image π m (cid:0) X δn ( f, U ) (cid:1) ⊂ L m ( X ) is endowed with an action of µ n given by λ.ϕ ( t ) = ϕ ( λt ), giving rise to aclass in M ˆ µX ( f ) ; we denote this motive by (cid:2) X δn,m ( f, U ) (cid:3) . Since X is smooth we have an equality (cid:2) X δn,m ( f, U ) (cid:3) L − md = (cid:2) X δn,n ( f, U ) (cid:3) L − nd ∈ M ˆ µX ( f ) . This element of M ˆ µX ( f ) is called the motivic measure of X δn ( f, U ) and denoted by mes (cid:0) X δn ( f, U ) (cid:1) .For any δ , consider the generating series Z δf,U ( T ) := X n ≥ mes (cid:0) X δn ( f, U ) (cid:1) T n ∈ M ˆ µX ( f ) [[ T ]] . By giving an explicit formula for this power series in terms of an embedded resolution of f and of F ,Guibert–Loeser–Merle show that it is rational, (see [ , Proposition 3.8]). One can then considerthe limit − lim T →∞ Z δf,U ( T ) as an element of M ˆ µX ( f ) . Moreover, they prove that the limit doesnot depend on the choice of δ , provided that it is big enough. This limit is the motive S f,U thatwe wanted to define. Remarks 2.1 . — 1. Assume that X is smooth and take U = X . Then in the constructionabove the condition on the tangency of arcs to F disappears, and there is therefore no needto show that the generating series does not depend on δ . The resulting motive S f := S f,X is the motivic nearby cycles defined by Denef–Loeser, and the procedure we sketched is theoriginal construction from [
8, 11 ]. It follows from the formula expressing S f in terms of alog-resolution of (cid:0) X, f − (0) (cid:1) that if 0 is not in the discriminant of f then S f = [ f − (0)].2. It follows from the construction of Guibert–Loeser–Merle that if X and F are smooth then S f (cid:0) [ i F : F → X ] (cid:1) = i F ! S f ◦ i F ,F = i F ! S f | F . In particular, by additivity of the morphism S f we obtain an equality S f,U = S f − i F ! S f | F .
3. Assume that k is the field of complex numbers. Bittner [ , §
8] and Guibert–Loeser–Merle[ , Proposition 3.17], extending a result of Denef–Loeser [ , Theorem 4.2.1], show that themotivic nearby cycles morphism is indeed a motivic version of the nearby cycles functor, OTIVIC AND ANALYTIC NEARBY FIBERS AT INFINITY AND BIFURCATION SETS which justifies the terminology. More precisely, they show that the following diagram iscommutative: M X S f / / χ X (cid:15) (cid:15) M ˆ µX ( f ) χ X f ) (cid:15) (cid:15) K (cid:0) D bc ( X ) (cid:1) ψ f / / K (cid:0) D bc ( X ( f )) mon (cid:1) , where K (cid:0) D bc ( X ) (cid:1) and K (cid:0) D bc ( X ( f )) mon (cid:1) are the Grothendieck rings of the derived cat-egories of bounded constructible complex sheaves on X and X ( f ) respectively (the latterbeing endowed with a quasi-unipotent action), ψ f is the nearby cycle functor at the level ofsheaves (see [ ]), and χ X and χ X ( f ) are the realization morphisms which assign to a variety p : Y → X or (cid:0) p : Y → X ( f ) , σ (cid:1) the corresponding class [ Rp ! Q Y ] and [ Rp ! Q Y , σ ] respectively.Moreover, the direct image functors at the level of motives and at the level of sheaves arecompatible. For instance, if i : U → X is an open immersion and f : X → A C is a morphism,then we have χ X ( f ) (cid:0) S f,U (cid:1) = h ψ f (cid:0) Ri ! Q U (cid:1)i and χ C (cid:0) f ! S f,U (cid:1) = h R f ! ψ f (cid:0) Ri ! Q U (cid:1)i .
3. Motivic nearby cycles at infinity and the motivic bifurcation set
In this section we study the motivic nearby cycles at infinity S ∞ f,a for a fiber f − ( a ) of a morphism f : U → A k , a motive introduced by the second author in [ ]. We prove that this object is a motivicgeneralization of the invariants λ a ( f ) mentioned in the introduction. We then deduce that under anatural assumption, that is when f has isolated singularities at infinity, the motivic bifurcation setof f , which is defined in loc. cit. as the set of those values a such that S ∞ f,a doesn’t vanish, containsthe topological bifurcation set of f . We begin by recalling some constructions of thesecond author from [ , § U be a smooth connected algebraic k -variety and let f : U → A k be a dominant morphism.By Nagata compactification theorem there exists a compactification ( X, i, ˆ f ) of f , by which wemean the data of a k -variety X , an open dominant immersion i : U → X , and a proper mapˆ f : X → A k such that the following diagram is commutative: U i / / f (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ X ˆ f (cid:15) (cid:15) A k In the following, we will identify U with its image i ( U ) in X , drop the immersion i from thenotation ( X, i, ˆ f ), and denote by F be the closed subvariety X \ U of X . LORENZO FANTINI & MICHEL RAIBAUT
Definition 3.1 (Motivic nearby cycles at infinity) . — For any a in A k ( k ), we call motivicnearby cycles at infinity of f for the value a the motive S ∞ f,a = ˆ f ! (cid:0) S ˆ f − a,U − i ! S f − a (cid:1) ∈ M ˆ µk . It is shown in [ , Theorem 4.2] that the motive S ∞ f,a does not depend on the chosen compact-ification of f . While the proof in loc. cit. relies on a computation on a suitable resolution ofthe singularities of the pair (cid:0) X, ˆ f − ( a ) ∪ F (cid:1) , this fact will also naturally follow from the resultsdiscussed in Section 4. Remark 3.2 . — The motive S ˆ f − a,U − i ! S f − a ∈ M ˆ µ ˆ f − ( a ) can be obtained using motivic integrationin the following way. Given integers δ > n ≥
1, consider the set of arcs X δ, ∞ n (cid:0) ˆ f − a (cid:1) = (cid:26) ϕ ∈ L ( X ) ϕ (0) ∈ F, ord F ( ϕ ) ≤ nδ ord (cid:0) ˆ f − a (cid:1) ( ϕ ) = n, ac (cid:0) ˆ f − a (cid:1) ( ϕ ) = 1 (cid:27) , and the zeta function Z δ, ∞ ˆ f − a,U ( T ) = X n ≥ mes (cid:0) X δ, ∞ n ( ˆ f − a ) (cid:1) T n ∈ M ˆ µ ˆ f − ( a ) [[ T ]] . Then, for any δ large enough, we have S ˆ f − a,U − i ! S f − a = − lim T →∞ Z δ, ∞ ˆ f − a,U ( T ) ∈ M ˆ µ ˆ f − ( a ) . The motive S ∞ f,a can be expected to be a motivic analogue of the invariants λ a ( f ) discussed inthe introduction. However, no link between the two was established in [ ]. That such a connectionexists is a consequence of the next Theorem. Theorem 3.3 . —
Let f be a polynomial in C [ x , . . . , x d ] and let a be an element of A C ( C ) . Then: χ c (cid:0) S ∞ f,a (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1) − χ c ( f ! S f − a ) ;2. if a is not a critical value of f , then χ c (cid:0) S ∞ f,a (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1) − χ c (cid:0) f − ( a ) (cid:1) ;3. if f has isolated singularities in A C , then χ c (cid:0) S ∞ f,a (cid:1) = ( − d − λ a ( f ) . Proof . — By applying the Euler characteristic to the definition of S ∞ f,a we obtain χ c (cid:0) ˆ f ! S ˆ f − a,U (cid:1) = χ c ( f ! S f − a ) + χ c (cid:0) S ∞ f,a (cid:1) , therefore we need to establish the following equality: χ c (cid:0) ˆ f ! S ˆ f − a,U (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1) . By [ , Proposition 3.17] (or [ , § a the motive ˆ f ! S ˆ f − a,U realizes on the class (cid:2) R ˆ f ! ψ ˆ f − a ( Ri ! Q U ) (cid:3) . Using the proper base change for the nearby cycles sheavesvia the application t − a in A k (see [ , Proposition 4.2.11]), we deduce that (cid:2) R ˆ f ! ψ ˆ f − a ( Ri ! Q U ) (cid:3) = (cid:2) ψ t − a (cid:0) ( R ˆ f ◦ Ri ) ! Q U (cid:1)(cid:3) = (cid:2) ψ t − a ( Rf ! Q U ) (cid:3) . Taking the Euler characteristics χ c for the motives and χ for the sheaves we obtain(3.1) χ c (cid:0) ˆ f ! S ˆ f − a,U (cid:1) = χ (cid:0) [ R ˆ f ! ψ ˆ f − a ( Ri ! Q U )] (cid:1) = χ (cid:0) [ ψ t − a ( Rf ! Q U )] (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1) , where the last equality follows from [ , (2.5.6)] and [ , (2.3.26)]. Hence we obtain χ c ( S ∞ f,a ) = χ c (cid:0) f − ( a gen ) (cid:1) − χ c ( f ! S f − a ) , proving part 1. OTIVIC AND ANALYTIC NEARBY FIBERS AT INFINITY AND BIFURCATION SETS In particular, if a ∈ A C ( C ) is not a critical value of f , we have f ! S f − a = [ f − ( a )], and therefore χ c (cid:0) f − ( a gen ) (cid:1) − χ c (cid:0) f − ( a ) (cid:1) = χ c (cid:0) S ∞ f,a (cid:1) , which proves part 2.Now assume that f has isolated singularities. Then the following equalities of motives over f − ( a ) hold: S f − a = (cid:2) f − ( a ) \ crit( f ) → f − ( a ) (cid:3) + X x ∈ f − ( a ) ∩ crit( f ) S f,x = (cid:2) f − ( a ) → f − ( a ) (cid:3) + X x ∈ f − ( a ) ∩ crit( f ) (cid:0) S f,x − (cid:2) x → f − ( a ) (cid:3)(cid:1) where, for any critical point x , the motive S f,x is the motivic Milnor fiber of f at the point x , whichis the motive constructed analogously as S f but only using arcs with origin x . This can be shownby subdividing the arcs defining S f − a according to whether their origin falls in the smooth part ofthe fiber or not, see [ , § f ! and taking the Euler characteristic of both sides of the last equation,we obtain χ c ( f ! S f − a ) = χ c (cid:0) f − ( a ) (cid:1) + X x ∈ f − ( a ) ∩ crit( f ) ( − d − µ ( f, x ) , where µ ( f, x ) is the Milnor number of f at x , since it follows from results of Denef–Loeser (forinstance from [ , Theorem 4.2.1]) that χ c ( S f,x −
1) = ( − d − µ ( f, x ).Finally, by applying to this the formula 1.2 and the previous part of the theorem we obtain theequality χ c (cid:0) S ∞ f,a (cid:1) = ( − d − λ a ( f ) , showing part 3 and thus concluding the proof of the theorem. In the light of the previous result, it is natural to define amotivic version of the topological bifurcation set of f as follows. Definition 3.4 ( [ , D´efinition 4.6] ) . — The motivic bifurcation set of f is the set defined as B mot f = (cid:8) a ∈ A k ( k ) (cid:12)(cid:12) S ∞ f,a = 0 (cid:9) ∪ disc( f ) , where disc( f ) denotes the discriminant of f .The set B mot f is finite, as is proven in [ , Th´eor`eme 4.13] via a computation on a suitableresolution of a compactification ( X, ˆ f ) of f .If we assume that k is the field of complex numbers, the first part of Theorem 3.3 implies thefollowing result about the motivic bifurcation set of f . Corollary 3.5 . —
Let f be a polynomial in C [ x , . . . , x d ] . Then we have (cid:8) a ∈ A C ( C ) (cid:12)(cid:12) χ c (cid:0) f − ( a ) (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1)(cid:9) ⊂ B mot f . Proof . — If a belongs to the discriminant of f then a belongs to B mot f by definition, therefore wecan assume that a is not in disc( f ). Now, since χ c (cid:0) f − ( a ) (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1) , it follows from thesecond part of Theorem 3.3 that χ c ( S ∞ f,a ) is nonzero, therefore S ∞ f,a is nonzero as well, that is a belongs to B mot f . LORENZO FANTINI & MICHEL RAIBAUT
Remark 3.6 . — In particular, whenever B top f = (cid:8) a ∈ A C ( C ) (cid:12)(cid:12) χ c (cid:0) f − ( a ) (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1)(cid:9) ,then B top f is included in B mot f . For example, as observed in the introduction, this holds in the caseof plane curves, or more generally whenever f has isolated singularities at infinity, by [ ]. Observealso that Theorem 3.3 does not require f to have isolated singularities in A d C , therefore it appliesalso to situations where the invariants λ a ( f ) are not defined. Remark 3.7 . — In this remark we explain how to lift the equality 1.3 to an equality in M ˆ µk . Inorder to do this we need to recall the notion of global motivic zeta function introduced in [ ]. Forany n ≥ δ ≥
1, we consider X global ,δn (cid:0) ˆ f (cid:1) = ϕ ( t ) ∈ L ( X ) ord F (cid:0) ϕ ( t ) (cid:1) ≤ nδ, ord (cid:16) ˆ f (cid:0) ϕ ( t ) (cid:1) − ˆ f (cid:0) ϕ (0) (cid:1)(cid:17) = n ac (cid:16) ˆ f (cid:0) ϕ ( t ) (cid:1) − ˆ f (cid:0) ϕ (0) (cid:1)(cid:17) = 1 . The global motivic zeta function , defined as Z global ,δ ˆ f,U ( T ) = X n ≥ mes (cid:0) X global ,δn,a ( ˆ f ) (cid:1) T n , is a rational series for δ large enough (see [ ][Theorem 4.10]), so that we can set S globalˆ f,U = − lim T →∞ Z global ,δ ˆ f,U ( T ) ∈ M ˆ µX . Moreover, again as in [ , Theorem 4.10], if f has finitely many critical points then by decom-posing the sets X global ,δn according to the origin of the arcs we obtain the following decomposition S globalˆ f,U = [ U \ crit( f ) → X ] + X x ∈ crit( f ) i ! S f,x + X a ∈ B mot f (cid:0) S ˆ f − a,U − i ! S f − a (cid:1) , and applying the direct image ˆ f ! we obtain the following equality in M ˆ µ A k :ˆ f ! S globalˆ f,U = (cid:2) U \ crit( f ) → A k (cid:3) + X x ∈ crit( f ) f ! S f,x + X a ∈ B mot f ˆ f ! (cid:0) S ˆ f − a,U − i ! S f − a (cid:1) . We claim that whenever k is the field of complex numbers this equality generalizes the equation1.3. Indeed, by pushing forward via the structure morphism p : A C → Spec C of A C and taking theEuler characteristic we obtain χ c (cid:16) p ! ˆ f ! S globalˆ f,U (cid:17) = 1 + ( − d − X x ∈ crit( f ) µ ( f, x ) + X a ∈ B mot f χ c (cid:0) S ∞ f,a (cid:1) . Now, observe that ˆ f ! S globalˆ f,U ∈ M ˆ µ A C is a motive over the affine line A C . Its fiber over a value a isthe motive ˆ f ! S ˆ f − a,U , hence all fibers have Euler characteristics equal to the Euler characteristic of f − ( a gen ) by the formula 3.1. Since the Euler characteristic of the base A C is equal to 1, we obtainthe equality χ c (cid:16) p ! ˆ f ! S globalˆ f,U (cid:17) = χ c (cid:0) f − ( a gen ) (cid:1) . Indeed, as in classical topology, if g : V → A is a surjective morphism whose fiber have all thesame Euler characteristic c then the Euler characteristic of V is c as well. This can be seen using OTIVIC AND ANALYTIC NEARBY FIBERS AT INFINITY AND BIFURCATION SETS constructible functions and integration against Euler characteristic (see for instance [ ] or [ ,Rem 4.1.32]) or more generally Cluckers–Loeser motivic integration [ ]. Thus, we obtain χ c (cid:0) f − ( a gen ) (cid:1) = 1 + ( − d − X x ∈ crit( f ) µ ( f, x ) + X a ∈ B mot f χ c (cid:0) S ∞ f,a (cid:1) , which, since χ c (cid:0) S ∞ f,a (cid:1) = ( − d − λ a ( f ) by part 3 of Theorem 3.3, yields χ (cid:0) f − ( a gen ) (cid:1) = 1 + ( − d − (cid:0) µ ( f ) + λ ( f ) (cid:1) , which is precisely the identity 1.3 cited in the introduction.
4. Analytic nearby fiber at infinity and the Serre bifurcation set
In this section, after recalling some constructions in non-archimedean geometry and motivicintegration, we define a non-archimedean analytic version of the nearby fiber at infinity and studyits properties.
We will briefly recall some basic notions of non-archimedean analytic geometry. While we chose to adopt the point of view of Berkovich, for thepurpose of this paper one could also work with rigid analytic spaces. We refer the reader to [ ]and to the references therein for a more thorough discussion of these theories.With any K -variety X is associated an analytic space over K , its analytification X an . It isa locally ringed space, whose points, in analogy with the theory of schemes, are the morphismsSpec K ′ → X , where K ′ is a valued field extension of K , modulo the relation that identifiestwo morphisms Spec K ′ → X and Spec K ′′ → X if they both factor through a third morphismSpec K ′′′ → X , where K ′′′ is an intermediate valued field extension of both K | K ′ and K | K ′′ .On the other hand, if X is a separated scheme of finite type over the valuation ring R of K , thenwe can also attach to it a formal scheme over R , its formal completion b X . As a locally ringed space, b X is isomorphic to the inverse limit of the schemes X ⊗ R R/ ( t n ); its underlying topological space is X k and its sheaf of functions is lim ←− O X ⊗ R R/ ( t n ) . The analytic space associated with b X (sometimesalso called the generic fiber of b X ), denoted by b X i , is the compact subspace of ( X K ) an consistingof those points x : Spec K ′ → X K that extend to an R -morphism ˜ x : Spec R ′ → X , where R ′ isthe valuation ring of K ′ . If such an extension exists then it is unique by the valuative criterionof separatedness, therefore we obtain a morphism sp b X : b X i → X k , called specialization , that isdefined by sending a point x as above to the image through ˜ x of the closed point of Spec R ′ . Remark 4.1 . — Let X be a separated scheme of finite type over R and let U be an open subschemeof X . Then the following results follows directly from the definitions:1. ( U K ) an is an open subspace of the K -analytic space ( X K ) an , and we have( X K ) an \ ( U K ) an = ( X K \ U K ) an .
2. If X is proper over R , then the inclusion of b X i in ( X K ) an is an isomorphism by the valuativecriterion of properness.3. The open immersion U → X induces an isomorphism b U i ∼ = sp − b X ( U ). Remark 4.2 . — Let X be a k -variety, let f : X → A k = Spec k [ t ] be a morphism, and set X R = X × A k Spec R and X K = X × A k Spec K . Then the k -arcs on X with origin in f − (0) correspond to LORENZO FANTINI & MICHEL RAIBAUT the totally ramified points of (cid:0)d X R (cid:1) i . Indeed, if ϕ : Spec k [[ s ]] → X is an arc on X with ord f ( ϕ ) = n >
0, then ϕ factors through X K and the compositionSpec k [[ s ]] ϕ −→ X K −→ Spec K identifies K ′ = Frac( k [[ s ]]) ∼ = K ( n ) with a totally ramified degree n extension of K , inducing a K ′ -point x of X an K whose specialization sp d X R ( x ) coincides with π ( ϕ ), hence a point of (cid:0)d X R (cid:1) i .Conversely, with a K ( n )-point of (cid:0)d X R (cid:1) i is associated a morphism Spec R ( d ) → X K , whose com-position with the projection X K → X is a k -arc on X such that ord f ( ϕ ) = n , as R ( n ) ∼ = k [[ t /n ]].For a more precise statement we refer the reader to [ , 6.1.2] or [ , 9.1.2].More generally, if X is a (separated and quasi-compact) formal R -scheme of finite type, that isa quasi-compact locally ringed space that is locally of the form b X for some separated R -schemeof finite type X , then by gluing the associated K -analytic spaces we obtain a compact K -analyticspace X i . We say that X is a formal model of X i , and we say that X is generically smooth if X i is a smooth K -analytic space. It follows from a celebrated theorem of Raynaud (see [ , § X is a K -variety then every compact subspace of X an admits a formal model; in particular,this is true for a distinguished class of compact subspaces of X an , its affinoid domains.For the purpose of motivic integration it is sufficient to have a formal model for the unramifiedpart of a given analytic space; this is formalized by the theory of weak N´eron models. If X is a(separated) K -analytic space, a weak N´eron model of X is a formal scheme of finite type X over R together with an open immersion i : X i → X such that for every unramified extension R ′ of R themap i induces a bijection X ( R ′ ) → X ( K ′ ), where K ′ is the fraction field of R ′ . Extending on workof Sebag [ ], Loeser–Sebag [ ] and Nicaise–Sebag [
25, 26, 27 ], Hartmann [ ] defined a theoryof equivariant motivic integration on formal schemes of finite type. We will briefly summarize theresults we need.Let n be an integer, let X be a generically smooth and flat formal R -scheme of finite type endowedwith a good µ n -action on X (meaning that every µ n -orbit is contained in an affine formal subschemeof X ), and let ω be a µ n -closed volume form on the compact K -analytic space X i (that is a nowherevanishing differential form of degree dim X i on X i satisfying an additional compatibility conditionin relation with the µ n -action, see [ , Definition 6.2, p.32]). Note that this setting includes thecase where there is no action, by taking n = 1; in which case the constructions below reduce tothose of [
33, 21, 26 ]. Then there exists a µ n -equivariant N´eron smoothening h : Y → X , that is anequivariant morphism of formal R -schemes such that Y is a weak N´eron model of X i and h factorsvia an open immersion through an equivariant morphism Y ′ → X that induces an isomorphism Y ′ i → X i . It can be shown that the motive Z X | ω | := X C connectedcomponent of Y k [ C ] L − ord C (cid:0) ( h i ) ∗ ( ω ) (cid:1) ∈ M µ n X k only depends on X and ω and not on the N´eron smoothening h .Moreover, if X is a smooth K -analytic space that admits a weak N´eron model U (such as forexample the space X i above, with U = Y ) and X is endowed with a µ n -action that extends to agood µ n -action on U , then the image of R U | ω | under the forgetful morphism M µ n U k → M µ n k only OTIVIC AND ANALYTIC NEARBY FIBERS AT INFINITY AND BIFURCATION SETS depends on X and ω and not on U , and is denoted by Z X | ω | ∈ M µ n k . Now let X be a generically smooth and flat formal R -scheme of finite type, and assume that X i admits a gauge form ω . Since weak N´eron models only see the unramified points of X i (which inthe setting of Remark 4.2 correspond only to arcs with contact order 1 along X k ), to see its ramifiedpoints we consider the volume Poincar´e series of ( X , ω ), that is defined as S ( X , ω, T ) = X n ≥ Z X ( n ) | ω ( n ) | T n ∈ M ˆ µ X k [[ T ]] , where for any integer n we denote by X ( n ) = X ⊗ R R ( n ) the base change of X to R ( n ) ∼ = k [[ t /n ]],endowed with the natural action of Gal( K ( n ) | K ) ≃ µ n . This series is rational, therefore it admitsa limit when T goes to infinity. The limit, which does not depend on the choice of ω by [ ,Proposition 8.1], is called the motivic volume of X and denoted byVol( X ) = − lim T →∞ S ( X , ω, T ) ∈ M ˆ µ X k . Given a smooth connected k -variety U endowed with a fixed volume form ω and a dominantmorphism f : U → A k , by taking a motivic volume of the compact K -analytic space (cid:0) c U R (cid:1) i onecan retrieve the motivic nearby cycles S f , see [ , Theorem 9.13] and [ , Proposition 7.7]. Moreprecisely, (cid:0) c U R (cid:1) i comes endowed with a canonical gauge form, its Gelfand–Leray form ω/df , andwe have Vol (cid:0) c U R (cid:1) = L − (dim U − S f ∈ M ˆ µU k . Its direct image by f ! does not depend on c U R , it is therefore called the motivic volume of (cid:0) c U R (cid:1) i ,denoted by Vol (cid:16)(cid:0) c U R (cid:1) i (cid:17) . We have(4.1) Vol (cid:16)(cid:0) c U R (cid:1) i (cid:17) = L − (dim U − f ! S f ∈ M ˆ µk . Let U be asmooth connected k -variety endowed with a fixed volume form ω , let f : U → A k be a dominantmorphism, let ( X, ˆ f ) be a compactification of f , and let F be the closed subset X \ U . After seeingthe results of the previous section one would expect to be able to retrieve the motive S ˆ f,U fromthe (non-compact) K -analytic space ( X K ) an , since the latter contains (cid:0) c U R (cid:1) i and all the pointscorresponding to the arcs on X that have origin in F but are not arcs in F . This is indeed the case,as shown by Bultot [ , § X, ˆ f ) of f is normal, that F is a Cartier divisor on X , and that we have additionaldata ( W α , g α ) α , where { W α } α is a finite cover of ( X K ) an by affinoid domains and, for each α , g α is an analytic function on W α such that W α ∩ ( F K ) an is defined by g α = 0 in W α . For everynonnegative integer γ , we set U γ := [ α ∈ A (cid:8) x ∈ W α (cid:12)(cid:12) | g α ( x ) | ≥ | t | γ (cid:9) . LORENZO FANTINI & MICHEL RAIBAUT
The U γ form an increasing sequence of compact analytic domains of ( U K ) an such that( U K ) an = [ γ ≥ U γ = [ α ∈ A (cid:8) x ∈ W α (cid:12)(cid:12) | g α ( x ) | > (cid:9) . In order to see that the motivic volumes of the U γ stabilize, it is useful to compute them relativelyto the formal scheme d X R in order to be able to compare them in the Grothendieck ring M ˆ µ X k . Thisis possible since, if U and U ′ are two weak N´eron models of U γ mapping to d X R , then there exists athird weak N´eron model of U γ dominating both, hence the images of the integral R U ω/df in M ˆ µ X k depends only on U γ and d X R , but not on the choice of U . In particular, the image of the motivicvolume Vol( U ) in M ˆ µ X k depends on U γ and not on U ; we denote this motive by Vol d X R ( U γ ).Bultot then proves that there exists an integer γ ≥ γ ≥ γ we haveVol( d X R )( U γ ) = Vol( d X R )( U γ ) ∈ M ˆ µX k . This motive does not depend on the sequence ( U γ ), we thus denote it by Vol d X R (cid:0) ( U K ) an (cid:1) andcall it motivic volume of ( U K ) an over d X R (mind that this motive is called motivic volume of U K over X k in [ ]). Furthermore, Bultot gives a formula computing Vol d X R (cid:0) ( U K ) an (cid:1) in terms of thecombinatorics of a good compactification of U , and by comparing it with the analogous formula ofGuibert–Loeser–Merle he shows that, as expected, we have(4.2) Vol d X R (cid:0) ( U K ) an (cid:1) = L − (dim U − S ˆ f,U ∈ M ˆ µX k . Since the direct image ˆ f ! S ˆ f,U in M ˆ µk does not depend on the choice of X , we can define the motivicvolume of ( U K ) an as Vol (cid:0) ( U K ) an (cid:1) = ˆ f ! (cid:0) Vol d X R (cid:0) ( U K ) an (cid:1)(cid:1) ∈ M ˆ µk . As before, let U be a smooth k -variety and let f : U → A k be a dominant morphism. We can now introduce non-archimedean analytic version ofthe motivic nearby fibers at infinity of f . Definition 4.3 . — Using these notations, we define the analytic nearby fiber at infinity of f forthe value K -analytic space F ∞ f, = ( U K ) an \ (cid:0) c U R (cid:1) i . Similarly, for every a in A k ( k ), we define the analytic nearby fiber at infinity of f for the value a to be the analytic nearby fiber at infinity of f − a for the value 0; we denote it by F ∞ f,a . Observethat these analytic spaces are clearly independent of the choice of a compactification of f . Remark 4.4 . — The definition of F ∞ f, is analogous to the one of S ∞ f, , for which one chooses acompactification ( X, ˆ f ) of ( U, f ) and considers only arcs with origin in F = X \ U . Indeed, aswe observed in Remark 4.2 the correct analogue for the origin of an arc on X is the specializa-tion sp d X R ( x ) of a point x of (cid:0)d X R (cid:1) i , and the inclusion of F ∞ f, in (cid:0)d X R (cid:1) i = ( X K ) an induces anisomorphism F ∞ f, ∼ = sp − d X R ( F ) \ ( F K ) an , since we have ( U K ) an ∼ = ( X K ) an \ ( F K ) an by the first part of Remark 4.1, and (cid:0) c U R (cid:1) i ∼ = sp − d X R ( U )by the third part of the same remark. Observe that removing ( F K ) an is necessary to obtain a space OTIVIC AND ANALYTIC NEARBY FIBERS AT INFINITY AND BIFURCATION SETS that does not depend on the choice of the compactification ( X, ˆ f ); this independence can also beseen as a consequence of the valuative criterion of properness, as two compactifications can alwaysbe dominated by a common third one.We declare the volume of F ∞ f, to beVol (cid:0) F ∞ f, (cid:1) = Vol (cid:0) ( U K ) an (cid:1) − Vol (cid:16)(cid:0) c U R (cid:1) i , (cid:17) ∈ M ˆ µk . Theorem 4.5 . —
Let U be a smooth connected k -variety and let f : U → A k be a dominant mor-phism. Then we have an equalityVol ( F ∞ f, ) = L − (dim U − S ∞ f, ∈ M ˆ µk Proof . — The result follows immediately from the definition of the volume and the equalities 4.1and 4.2.
Remark 4.6 . — Another approach to motivic integration was developed by Hrushovski–Kazhdan[ ] with model theoretic methods and gives a group morphism Vol HK from the Grothendieck ringof semi-algebraic sets over the valued field K to M ˆ µk (see also [ , Theorem 2.5.1]). In particu-lar, the volume of any locally closed subset of ( U K ) an is defined, and the equality Vol HK ( F ∞ f, ) =Vol HK (cid:0) ( U K ) an (cid:1) − Vol HK (cid:0)(cid:0) c U R (cid:1) i (cid:1) holds naturally in this context. Moreover, the motivic volumesin [ ] can be computed in an analogous way as the volumes in [ ] and [ ] in terms of suit-able resolutions of singularities (see [ , Theorem 2.6.1]), and this can be used to show thatVol HK (cid:0)(cid:0) c U R (cid:1) i (cid:1) = f ! S f , see [ , Corollary 2.6.2] or [ ]. One should similarly be able to de-duce that Vol HK (cid:0) ( U K ) an (cid:1) = ˆ f ! S ˆ f,U , so that Vol HK ( F ∞ f, ) is the volume Vol( F ∞ f, ) that we definedabove. Observe that, since our definition of the analytic nearby fiber at infinity is independent ofthe choice of a compactification of f , as is the morphism Vol HK , this approach would also yield acompactification-independent definition of the motivic nearby cyles at infinity. We will now recall the notion of motivic Serre invariant and useit to define another bifurcation set B ser f which we call the Serre bifurcation set of f .Since the Euler characteristic is additive on the category of k -varieties, it gives rise to a morphism χ c : M ˆ µk → Z . Moreover, since χ c ( L −
1) = χ c ( G m,k ) = 0, this morphism factors through amorphism χ c : M ˆ µk / ( L − −→ Z . Therefore, when interested in working with the Euler characteristic it is often useful to study theclass of a variety modulo L −
1. Now, the constructions made in this section can all be done modulo L −
1, which leads to some simplifications. Indeed, if X is a generically smooth and flat formal R -scheme of finite type endowed with a µ n -action, ω is a µ n -closed gauge form on X i , and h : Y → X is a µ n -equivariant N´eron smoothening, then in M ˆ µk we have Z X | ω | ≡ X C connectedcomponent of Y k [ C ] L − ord C (cid:0) ( h i ) ∗ ( ω ) (cid:1) ≡ X C [ C ] = [ Y k ] mod ( L − . This element of M ˆ µk / ( L − X and not on ω , is called the motivic Serreinvariant of X , and denoted by S ( X ). LORENZO FANTINI & MICHEL RAIBAUT
Remark 4.7 . — The motivic Serre invariant of a generically smooth formal R -scheme of finitetype was introduced by Loeser–Sebag [ ], and developed by Nicaise–Sebag [
26, 25, 27 ], Bultot[ ] and Hartmann [ ]. It generalizes an invariant introduced by Serre in [ ] in order to classifythe compact analytic manifolds over a local field L and defined using classical p -adic integrationwith value in the ring Z / ( q − Z , where q is the cardinality of the residue field l of L . Counting l -rational points yields a canonical morphism M l / ( L − → Z / ( q − Z , and Loeser–Sebag showedthat the image by this morphism of the motivic Serre invariant S ( X ) of a smooth and compact L -analytic space X is equal to the classical Serre invariant of the underlying compact manifold.We then consider the motive(4.3) S ( X ) := Vol( X ) mod ( L − ∈ M ˆ µk / ( L − , which can also be obtained as the limit of the generating series − P n ≥ S ( X ( n )) T n . Similarly, if X is a smooth K -analytic space admitting a weak N´eron model U over R , one also sets S ( X ) = S ( U ),and we obtain a motive(4.4) S ( X ) = Vol( X ) mod ( L − ∈ M ˆ µk / ( L − U, X, ˆ f, { U γ } γ are as in subsection 4.3, then S (cid:0) ( U K ) an (cid:1) := S ( U γ ) ∈ M ˆ µk / ( L − U and f and not on γ , if γ is large enough, nor on X , { U γ } γ , and ˆ f . We can thendefine the Serre invariant of F ∞ f, as S ( F ∞ f, ) := S (cid:0) ( U K ) an (cid:1) − S (cid:0) c U R (cid:1) ∈ M ˆ µk / ( L − . Definition 4.8 . — Let U be a smooth k -variety and let f : U → A k be a dominant morphism.The Serre bifurcation set of f is B ser f = (cid:8) a ∈ A C (cid:12)(cid:12) S (cid:0) F ∞ f,a (cid:1) = 0 (cid:9) ∪ disc( f ) . Theorem 4.9 . —
Let U be a smooth k -variety and let f : U → A k be a dominant morphism. Thenwe have an equality S ( F ∞ f, ) = S ∞ f, mod ( L − in M ˆ µk / ( L − , and B ser f is contained in B mot f .Proof . — By combining the definition of S ( F ∞ f, ) and 4.3 and 4.4 with the formulas 4.1 and 4.2, wededuce that the two equalities S (cid:0) ( U K ) an (cid:1) = L − (dim U − ˆ f ! (cid:16) S ˆ f,U (cid:17) mod ( L − S (cid:16) c U R (cid:17) = L − (dim U − f ! ( S f ) mod ( L − M ˆ µk / ( L − Corollary 4.10 . —
Let f be a polynomial in C [ x , . . . , x d ] . Then we have the inclusions (cid:8) a ∈ A C ( C ) (cid:12)(cid:12) χ c (cid:0) f − ( a ) (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1)(cid:9) ⊂ B ser f ⊂ B mot f . Proof . — The result follows from from Corollary 3.5, together with the fact that for any value a the Euler characteristic χ c (cid:0) S ( F ∞ f,a ) (cid:1) and χ c (cid:0) S ∞ f,a (cid:1) are equal. OTIVIC AND ANALYTIC NEARBY FIBERS AT INFINITY AND BIFURCATION SETS Remark 4.11 . — In particular, whenever B top f = (cid:8) a ∈ A C ( C ) (cid:12)(cid:12) χ c (cid:0) f − ( a ) (cid:1) = χ c (cid:0) f − ( a gen ) (cid:1)(cid:9) , asin the case of plane curves, or more generally whenever f has isolated singularities at infinity, wehave a chain of inclusions B top f ⊂ B ser f ⊂ B mot f . References [1]
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Lorenzo Fantini , Aix-Marseille Universit´e, Institut de Math´ematiques de Marseille, 13453 Marseille, France
E-mail : [email protected] • Url :
Michel Raibaut , Laboratoire de Math´ematiques, Univ. Grenoble Alpes, Universit´e Savoie Mont Blanc,Bˆatiment Chablais, Campus Scientifique, Le Bourget du Lac, 73376 Cedex, France