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J reine angew Math., Ahead of Print DOI 10.1515 / crelle-2015-0022 Journal für die reine und angewandte Mathematik © De Gruyter 2015 The motivic Thom–Sebastiani theorem for regular and formal functions By Quy Thuong Lê at Hanoi Abstract Thanks to the work of Hrushovski and Loeser on motivic Milnor fibers, we give a model-theoretic proof for the motivic Thom–Sebastiani theorem in the case of regular functions Moreover, slightly extending Hrushovski–Loeser’s construction adjusted to Sebag, Loeser and Nicaise’s motivic integration for formal schemes and rigid varieties, we formulate and prove an analogous result for formal functions The latter is meaningful as it has been a crucial element of constructing Kontsevich–Soibelman’s theory of motivic Donaldson–Thomas invariants Introduction Let f and g be holomorphic functions on complex manifolds of dimensions d1 and d2 , having isolated singularities at x and y, respectively Define f ˚ g by f ˚ g.x; y/ D f x/ C g.y/: Let Ff;x be the (topological) Milnor fiber of f; x/, the same for g; y/ and f ˚ g; x; y// The original Thom–Sebastiani theorem [24] states that there exists an isomorphism between the cohomology groups H d1 Cd2 Ff ˚g;.x;y/ ; Q/ Š H d1 Ff;x ; Q/ ˝ H d2 Fg;y ; Q/ compatible with the monodromies Steenbrink in [25] refined a conjecture on the Thom– Sebastiani theorem for the mixed Hodge structures, which was fulfilled later and independently by Varchenko [27] and Saito [22] In the letters to A’Campo (1972) and to Illusie (1999), Pierre Deligne discussed the `-adic version for an arbitrary field (rather than complex numbers), in which he replaced the Milnor fibers by the nearby cycles and used Laumon’s construction of convolution product (cf [16, Définition 2.7.2]); this work recently has been fully realized by The author is partially supported by the Centre Henri Lebesgue (program “Investissements d’avenir”, ANR11-LABX-0020-01) and by ERC under the European Community’s Seventh Framework Programme (FP7/20072013), ERC Grant Agreement no 246903/NMNAG Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM Lê, Regular and formal motivic Thom–Sebastiani theorem Fu [8] Furthermore, Denef–Loeser [5] and Looijenga [19] also provided proofs of the motivic version for motivic vanishing cycles in the case of fields of characteristic zero, from which the classical results were recovered without the hypothesis that x and y are isolated singularities We come back to the problem on the motivic Thom–Sebastiani theorem in the framework for the motivic Milnor fibers of formal functions It has been likely a formally unsolved problem, but already used in Kontsevich–Soibelman’s theory of motivic Donaldson–Thomas invariants for non-commutative Calabi–Yau threefolds (see [15]) Using Temkin’s results on resolution of singularities of an excellent formal scheme [26] and Denef–Loeser’s formulas for the motivic Milnor fiber of a regular function [3, 6], Kontsevich and Soibelman introduce in [15] the motivic Milnor fiber of a formal function The motivic Thom–Sebastiani theorem for formal functions that concerns this notion is a key to construct the motivic Donaldson–Thomas invariants In fact, it has the same interpretation as Denef–Loeser’s and Looijenga’s local version (cf [5, 19]) and a complete proof for it should be required This is the main purpose of the present article The motivic Milnor fiber of a regular function may be described in terms of resolution of singularity, after the works of Denef–Loeser [3, 6, 7] and of Guibert–Loeser–Merle [10– 12] In particular, Guibert–Loeser–Merle had the refinement when applying this method to further extensions of the motivic Thom–Sebastiani theorem (see [10–12]) Recently, with the help of Hrushovski–Kazhdan’s motivic integration, Hrushovski and Loeser [14] have given an even more flexible manner to describe the motivic Milnor fiber in terms of the data of the corresponding analytic Milnor fiber (introduced by Nicaise–Sebag [21]) An important application of this approach is our proof of the integral identity conjecture in [17] There it is also shown that a slight generalization of Hrushovski–Loeser’s construction [14] combined with Nicaise’s formula on volume Poincaré series [20] allows to interpret the motivic Milnor fiber of a formal function in the same way as in [14] However, this method requires the restriction to studying the motivic Milnor fibers over algebraically closed fields of characteristic zero (hence the hypothesis in the present work) Our article is organized as follows In Section 2, we recall some basic and essential backgrounds on the motivic Milnor fiber of a regular function, in which the local form of Denef–Loeser’s and Looijenga’s motivic Thom–Sebastiani theorem is included (Theorem 2.1), using the main references [3–7] and [19] The local form states that Sf ˚g;.x;y/ D Sf;x Sg;y ; where Sf;x is the motivic Milnor fiber of f; x/, Sf;x WD 1/d1 Sf;x 1/, the same for g; y/ and f ˚g; x; y//, and is the convolution product (cf Section 2.3) Here, one does not need to assume that x and y are isolated singular points Using the tools from [13,14], recalled partly in Section below, we introduce a new proof for this formula in Section Notice that the previous formula lives in the monodromic Grothendieck ring MkO , by a technical reason, however, our proof only runs in a localization of MkO In Section 3, we mark the highlights and the essences of motivic integration for special formal schemes, following [17, 18, 20, 21, 23] In particular, by [17], we show that Kontsevich– Soibelman’s motivic Milnor fiber of a formal function and Nicaise’s volume Poincaré series mention the same thing, which can be also read off from the corresponding analytic Milnor fiber Furthermore, we can use the model-theoretic tools recalled in Section to describe the volume Poincaré series, hence the motivic Milnor fiber of a formal function The formal Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM Lê, Regular and formal motivic Thom–Sebastiani theorem version of the motivic Thom–Sebastiani theorem has the same form as the regular one but f and g replaced by formal functions f and g, respectively (Theorem 3.4) This is proved in Section using the development of tools in Section as well as some analogous techniques in the proof of the regular version in Section Preliminaries Throughout the present article, we always assume that k is an algebraically closed field of characteristic zero 2.1 Grothendieck rings of algebraic varieties By definition, an algebraic k-variety is a separated reduced k-scheme of finite type Let Vark be the category of algebraic k-varieties, its morphisms are morphisms of algebraic k-varieties The Grothendieck group K0 Vark / is an abelian group generated by symbols ŒX for objects X in Vark subject to the relations ŒX D ŒY if X and Y are isomorphic in Vark , ŒX D ŒY C ŒX n Y if Y is Zariski closed in X Moreover, K0 Vark / is also a ring with unit with respect to the cartesian product Set L WD ŒA1k and denote by Mk the localization of K0 Vark / with respect to the multiplicative system ¹Li j i Nº Let m (or m k/) be the group scheme of mth roots of unity in k Varying m in N, such schemes give rise to a projective system with respect to morphisms mn ! m given by 7! n , and its limit will be denoted by O A good m -action on an object X of Vark is a group action of m on X such that each orbit is contained in an affine k-subvariety of X A good O -action on X is a O -action which factors through a good m -action for some m in N The O -equivariant Grothendieck group K0O Vark / is an abelian group generated by the iso-equivariant classes of varieties ŒX; , with X an algebraic k-variety, a good O -action on X, modulo the conditions ŒX; D ŒY; jY C ŒX n Y; jXnY if Y is Zariski closed in X and ŒX Ank ; D ŒX Ank ; if , lift the same O -action on X to an affine action on X Ank In the present article we shall denote ŒX; simply by ŒX when the O -action is clear Similarly, K0O Vark / has a natural ring structure due to the cartesian product Let O MkO denote K0O Vark /ŒL , it is the O -equivariant version of Mk above Let Mk;loc be the O localization of Mk with respect to the multiplicative family generated by the elements Li , O with i in N We shall also write loc for the localization morphism MkO ! Mk;loc 2.2 Motivic Milnor fiber Let X be a pure d -dimensional smooth k-variety, f a nonconstant regular function on X, and x a closed point in the zero locus of f Denote by Xx;m (or Xx;m f /) the set of arcs '.t / in X.kŒt =.t mC1 // originated at x with f '.t // Á t m mod t mC1 , which is a locally closed subvariety of k-variety X.kŒt =.t mC1 // Since Xx;m is invariant by the good m -action on X.kŒt =.t mC1 // given by '.t / D ' t /, it defines an iso-equivariant class ŒXx;m in MkO The motivic zeta function of f at x is the formal series X Zf;x T / D ŒXx;m L md T m m with coefficients in MkO Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM Lê, Regular and formal motivic Thom–Sebastiani theorem By Denef–Loeser [3], Zf;x T / is a rational function, i.e., an MkO -linear combination of and finite products (possibly empty) of La T b =.1 La T b / with a; b/ in Z N>0 Remark that we can take by [6] the limit limT !1 for rational functions such that La T b D T !1 La T b lim Then the motivic Milnor fiber of f at x is defined as This is a virtual variety in MkO 1: limT !1 Zf;x T / and denoted by Sf;x 2.3 The motivic Thom–Sebastiani theorem for regular functions In this subsection, we restate the motivic Thom–Sebastiani theorem for motivic Milnor fibers Let us recall from [5, 11, 19] the concept of convolution product Consider the Fermat defined by the equations um C v m D and um C v m D 1, varieties F0m and F1m in Gm;k respectively We endow with the standard m m /-action on these varieties If X and Y are algebraic k-varieties with m -action, one defines ŒX ŒY D ŒF1m m m X Y/ C ŒF0m m m X Y/; where, for i ¹0; 1º, Fim m m X Y/ D Fim X Y/= with au; bv; x; y/ u; v; ax; by/ for any a, b in m The group scheme m acts diagonally m m X on Fim Y/ Passing to the projective limit that MkO equals limMk m , we get the convolution product on MkO This product is commutative and associative (see for example [11]) Let f and g be regular functions on smooth algebraic k-varieties X and Y, respectively Define f ˚ g.x; y/ D f x/ C g.y/ For closed points x in X0 and y in Y0 , we set Sf;x D 1/dim X Sf;x 1/; Sg;y D 1/dim Y Sg;y 1/ and similarly for Sf ˚g;.x;y/ Theorem 2.1 ([5, 19]) The identity Sf ˚g;.x;y/ D Sf;x Sg;y holds in MkO Remark 2.2 In fact, in [5] and [19], the motivic Thom–Sebastiani theorem is proved in the framework of motivic vanishing cycles, which implies Theorem 2.1 The motivic Thom–Sebastiani formula for formal functions Let X be a generically smooth special formal kŒŒt -scheme of relative dimension d , with reduction X0 and structural morphism f Let x be a closed point of X0 3.1 The motivic Milnor fiber of a formal function By [26] (see also [20]), there exists a resolution of singularities h W Y ! X of X0 Let Ei , i J , be the irreducible Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM Lê, Regular and formal motivic Thom–Sebastiani theorem components of Ys /red Let Ni be the multiplicity of Ei in Ys We set Ei D Ei /0 for i J , and \ [ EI D Ei and EIı D EI n Ej i 2I j 62I for a nonempty subset I of J Let ¹U º be a covering of Y by affine open subschemes with U \ EIı 6D ; such that, on this piece, Y N f ı h D uQ yi i ; i 2I where uQ is a unit, yi is a local coordinate defining Ei Set mI WD gcd.Ni /i 2I One can construct as in [7] an unramified Galois covering I W EQIı ! EIı with Galois group mI , which is given over U \ EIı by ® ¯ z; y/ A1k U \ EIı / j z mI D u.y/ Q : Note that EQIı is endowed with a natural mI -action good over EIı obtained by multiplying the z-coordinate with elements of mI We also restrict this covering over EIı \ h x/ and obtain a class, written as ŒEQIı \ h x/, in MkO The motivic Milnor fiber of the formal germ X; x/, or of f at x, is defined to be the quantity X L/jI j ŒEQIı \ h x/ ;6DI J in MkO We denote it by S.X; x/ or by Sf;x By [17, Lemma 5.7], using volume Poincaré series, Sf;x is well defined, i.e., independent of the choice of the resolution of singularities h Remark 3.1 Let b Xx denote the formal completion of X at x, and let fx be the structural morphism of b Xx , which is induced by f We are able to use a resolution of singularity of X at x to define the motivic Milnor fiber Sfx ;x Then, it is clear that Sf;x D Sfx ;x 3.2 Integral of a gauge form and volume Poincaré series Stft formal schemes Assume that X is a separated generically smooth formal kŒŒt scheme topologically of finite type and that the relative dimension of X is d One may regard X as the inductive limit of the kŒt =.t mC1 /-schemes locally of finite type Xm WD X; OX ˝kŒŒt kŒt =.t mC1 / in the category of formal kŒŒt -schemes By Greenberg [9], there exists a unique k-scheme Grm Xm / topologically of finite type, up to isomorphism, which for any k-scheme Y admits a natural bijection Homk Y; Grm Xm // ! HomSpec.k/ Y k kŒt =.t mC1 /; Xm : These k-schemes Grm Xm / together with the natural translation give rise to a projective system, we denote its limit by Gr.X/ (cf [18, 23]) We denote by m the canonical projection Gr.X/ ! Grm Xm / See more in [9] for some basic properties of the functor Gr Notice that the notion of stable cylinder of Gr.X/ in this context was already introduced in [18, 23] Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM Lê, Regular and formal motivic Thom–Sebastiani theorem By [18, 23], the motivic measure of a stable cylinder A in Gr.X/ is A/ D Œ (3.1) ` A/L `C1/d for ` N large enough Let ˛ W A ! Z [ ¹1º be a function on A that takes only a finite number of values such that every fiber ˛ m/ is a stable cylinder in Gr.X/ Let ! be a gauge form on XÁ By [2, Proposition 1.5] (see also [18]), there exists a canonical isomorphism d XÁ XÁ / Š d XjkŒŒt X/ ˝kŒŒt k t //; Q Let ' thus there exist an n in N and a differential form !Q in dXjkŒŒt X/ such that ! D t n ! be a point of Gr.X/ outside Gr.Xsing / Then, we can regard it as a morphism of formal schemes Spf.kŒŒt / ! X, or as a morphism of rings OX X/ ! kŒŒt Thus it induces a morphism of rings 'Q W ' dXjkŒŒt X/ ! kŒŒt , which is a surjection One defines ord.!/.'/ Q D ord t '.' Q !// Q and (3.2) ordX !/ D ord.!/ Q n: The latter is independent of the choice of !Q (cf [18]) Since ! is a gauge form, it follows from the proof of [18, Theorem-Definition 4.1.2] that ordX !/ is an integer-valued function taking only a finite number of values and that its fibers are stable cylinder Then one defines (cf [18, 23]) Z X ® ¯ j!j WD ' Gr.X/ j ordX !/.'/ D m L m Mk : XÁ m2Z Special formal schemes We consider the more general case where X is a generically smooth special formal kŒŒt -scheme (see [1] for definition) Let Y ! X be a Néron smoothening for X, i.e., a morphism of special formal kŒŒt -schemes, with Y being adic smooth over kŒŒt, inducing an open embedding YÁ ! XÁ with b k t // K D XÁ ˝ b k t // K YÁ ˝ for any finite unramified extension K of k t // Such a Néron smoothening exists by [20] Furthermore, we are able to (and we shall from now on) choose Y to be a separated generically smooth formal kŒŒt -scheme topologically of finite type Using [20, Propositions 4.7, 4.8], for any gauge form ! on XÁ , we define Z Z j!j WD j!j Mk : XÁ YÁ For any m in N>0 , let b kŒŒt kŒŒt 1=m ; X.m/ WD X ˝ b k t // k t 1=m //; XÁ m/ WD XÁ ˝ and let !.m/ be the pullback of ! via the natural morphism XÁ m/ ! XÁ The Néron smoothening Y ! X for X induces a Néron smoothening Y.m/ ! X.m/ for X.m/, and Y.m/ is also topologically of finite type, like Y The canonical -action on Gr.Y.m/ is R 1=m / D '.at 1=m / It induces a given by a'.t -action on j!.m/j, thus we regard m XÁ m/ R O XÁ m/ j!.m/j as an element of Mk Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM Lê, Regular and formal motivic Thom–Sebastiani theorem Volume Poincaré series Let X be a generically smooth special formal kŒŒt -scheme, x a closed point of X0 , and b Xx the formal completion of X at x Denoting by xŒ the tube of x, namely the analytic Milnor fiber of f at x (cf [21]), we have the canonical isomorphism xŒ Š b Xx /Á Set xŒm WD xŒ k t // k t 1=m //: Let us consider the volume Poincaré series of xŒ; !/, where ! is a gauge form on xŒ (cf [20]): à X ÂZ j!.m/j T m MkO ŒŒT : S.xŒ; !I T / WD m xŒm Remark 3.2 More generally, the volume Poincaré series of separated generically smooth formal schemes topologically of finite type (resp separated quasi-compact smooth rigid varieties) were introduced and studied first by Nicaise–Sebag in [21] After that, Nicaise [20] studied these objects in the framework of generically smooth special formal schemes (resp bounded smooth rigid varieties) In practice, one may assume that ! is b Xx -bounded, i.e., ! lies in the image of the natural map (cf [20, Definition 2.11]) d b Xx jkŒŒt ˝kŒŒt k t // b Xx / ! d xŒjk t // xŒ/: Since k is an algebraically closed field, S.xŒ; !I T / is independent of the choice of the uniformizing parameter t Indeed, let t be another uniformizing parameter for kŒŒt Then t D ˛t, where ˛ D ˛.t / kŒŒt and ˛.0/ k Since k contains all roots, the mth roots of ˛ are again in kŒŒt This induces a canonical isomorphism of k t //-fields k t 1=m // ! k t 01=m //; which implies the previous claim By Nicaise [20, Corollary 7.13], if the gauge form ! is b Xx bounded, this series S.xŒ; !I T / is a rational function Proposition 3.3 With the notation and the hypotheses as previous, the following identity holds in MkO : à X ÂZ d Sf;x D L lim j!.m/j T m : T !1 m xŒm Proof The identity is true in Mk because of the definition of Sf;x as well as Nicaise’s formula for limT !1 S.xŒ; !I T / in [20, Proposition 7.36] To see that it is true in MkO , we refer to the proof of [17, Lemma 5.7] 3.3 Statement of result for formal functions Let d1 ; d2 be integers with d1 and d2 Let f be a formal power series in kŒŒx with f 0/ D and g in kŒŒy with g.0/ D Here x D x1 ; : : : ; xd1 /, y D y1 ; : : : ; yd2 / and we use the same symbol for the origin of Adk1 , Adk2 or A1k (whenever necessary, e.g., in Section 6, however, we shall write 0di for the origin of Adki , i ¹1; 2º) Let us consider the following special formal kŒŒt -schemes: Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM Lê, Regular and formal motivic Thom–Sebastiani theorem X WD Spf kŒŒt; x=.f x/ t/ ; Y WD Spf kŒŒt; y=.g.y/ t/ ; X ˚ Y WD Spf kŒŒt; x; y=.f x/ C g.y/ t/ ; with structural morphisms f, g and f ˚ g induced by f , g and f ˚ g, respectively Set Sf;0 WD 1/d1 Sf;0 1/ for f and the same for g and f ˚g We now set up the statement of the motivic Thom–Sebastiani O , using Hrushovski– theorem for formal schemes and then prove it in the setting of Mk;loc Kazhdan’s integration [13] via the work of Hrushovski–Loeser [14] Theorem 3.4 The identity loc.Sf˚g;.0;0/ / D loc.Sf;0 O Sg;0 / holds in Mk;loc The complete proof is given in Section Extension of Hrushovski–Loeser’s morphism 4.1 The theory ACVFk t// 0; 0/ Let us consider the theory ACVFk t // 0; 0/ of algebraically closed valued fields of equal characteristic zero that extend k t // (cf [13]) Its sort VF admits the language of rings, while the sort RV is endowed with abelian group operations and =, a unary predicate k for a subgroup, and a binary operation C on k D k [ ¹0º We also have an imaginary sort that is with a uniquely divisible abelian group For a model L of this theory, let RL (resp mL ) denote its valuation ring (resp the maximal ideal of RL ) The following are the “elementary” L-definable sets of ACVFk t // 0; 0/: VF.L/ D L; RV.L/ D L =.1 C mL /; .L/ D L =RL ; k.L/ D RL =mL : In general, a definable subset of VFn L/ is a finite Boolean combination of set of the forms val.f1 / Ä val.f2 / or f3 D 0, where fi are polynomials with coefficients in k t // The same definition may apply to definable subsets of RVn L/, n L/ or kn L/ Correspondingly, there are the following natural maps between these sets: rv W VF ! RV; val W VF ! ; valrv W RV ! ; res W RL ! k.L/: There is an exact sequence of groups: ! k ! RV valrv ! ! 0: 4.2 Measured categories (following [13]) VF-categories Let VF be the category of k t //-definable sets (or definable sets, for short) endowed with definable volume forms, up to -equivalence One may show that it is graded via the following subcategories VFŒn, n N An object of VFŒn is a triple X; f; "/ with X a definable subset of VF` RV` , for some `, `0 in N, f W X ! VFn a definable map with finite fibers, and " W X ! a definable function; a morphism from X; f; "/ Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM Lê, Regular and formal motivic Thom–Sebastiani theorem to X ; f ; "0 / is a definable essential bijection F W X ! X such that " D "0 ı F C val.JacF / away from a proper closed subvariety of X (the measure preserving property) Here, that F W X ! X is an essential bijection means that there exists a proper closed subvariety Y of X such that F jXnY W X n Y ! X n F Y / is a bijection (see [13, Section 3.8]) Let VFbdd Œn be the full subcategory of VFŒn whose objects are bounded definable sets with bounded definable forms " If considering " W X ! as the zero function, we obtain the categories volVF and volVFŒn as well as volVFbdd and volVFbdd Œn In this case, the measure preserving property of a morphism F is characterized by the condition val.JacF / D 0, outside a proper closed subvariety Convention For simplicity, we shall omit the symbol f in the triple X; f; "/ when no confusion can arise RV-categories Similarly, we consider the category RV graded by RVŒn, n N By definition, an object of RVŒn is a triple X; f; "/ with X a definable subset of RV` , for some ` in N, f W X ! RVn a definable map with finite fibers, and " W X ! a definable function; a morphism X; f; "/ ! X ; f ; "0 / is a definable bijection F W X ! X such that " C jvalrv f /j D "0 ı F C jvalrv f ı F /j away from a proper closed subvariety Here, for x D x1 ; : : : ; xn / n , we define jxj as the P sum niD1 xi The category RESŒn is defined as the full subcategory of RVŒn such that, for each object X; f; "/, valrv X / is a finite set The category RVbdd is defined as RV with valrv -image of objects bounded below In the case where, for each object X; f; "/ of one of the previous categories, " is the zero function, we obtain the subcategories volRV, volRVbdd and volRES In the present article, we also consider RES, a category defined exactly as volRES but the measure preserving property is not required for morphisms -categories The category Œn consists of pairs ; l/ with a definable subset of n and l W ! a definable map A morphism ; l/ ! 0 ; l / is a definable bijection W ! 0 which is liftable to a definable bijection valrv1 ! valrv1 0 such that jxj C l.x/ D j x/j C l x//: The category bdd Œn is the full subcategory of Œn such that, for each object ; l/ of bdd Œn, there exists a with Œ ; 1/n By definition, the categories and bdd L L are the direct sums n Œn and n bdd Œn, respectively The subcategories whose objects are of the form ; 0/ will be denoted by vol and vol bdd 4.3 Structure of K VFbdd / Let C be one of the categories in Section 4.2 Then, as in [13], we denote the Grothendieck semiring of C by KC C/ and the associated ring by K.C/ By [13], there is a natural morphism of semirings (4.1) N W KC bdd / ˝ KC RES/ ! KC bdd / VF Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM 10 Lê, Regular and formal motivic Thom–Sebastiani theorem constructed as follows Note that two objects admitting a morphism in bdd Œn define the same element in KC bdd Œn/, hence lifts to a morphism in VFbdd Œn between their pullbacks Thus there exists a natural morphism KC bdd Œn/ ! KC VFbdd / mapping the class of ; l/ to the class of val /; l ı val/ Also, for each object X; f; "/ in RESŒn, we may consider an étale map ` W X ! kn By this, we have the natural morphism KC RESŒn/ ! KC bdd / VF by sending the class of X; f; "/ to the class of X `;res Rn ; pr1 ı "/ In particular, if X is Zariski open in kn , then X `;res Rn is simply res X / Theorem 4.1 (Hrushovski–Kazhdan [13]) The morphism N is a surjection Moreover, it also induces a surjective morphism N between the associated rings There is a more intrinsic description of N , which follows from [13, Theorem 8.29, Proposition 10.10] More precisely, one first constructs the natural morphism KC bdd / ˝ KC RES/ ! KC bdd / RV due to the inclusion RES RV and the valuation map valrv This morphism is a surjection, its kernel is generated by ˝ Œvalrv1 /1 Œ 1 ˝ 1, with definable in The subscript means that the classes are in degree Second, the canonical morphism KC bdd Œn/ VF ! KC bdd Œn/=Œ11 ŒRV>0 1 RV induced by the map Ob RVŒn ! Ob VFŒn sending X; f; "/ to LX; Lf; L"/, where LX D X f;rv VF /n , Lf a; b/ D f a; rv.b// and L".a; b/ D ".a; rv.b//, is an isomorphism Then the composition of the first morphism with the natural projection KC bdd Œn/ RV ! KC bdd Œn/=Œ11 ŒRV>0 1 RV and with the inverse morphism of the second morphism (with all n) yields the morphism N Remark 4.2 According to [13, Proposition 10.10], an element of KC RVbdd / may be written as a finite sum of elements of the form Œ.X valrv1 /; f; "/ Furthermore, an argument in the proof of [13, Proposition 10.10] implies Œ.X valrv1 /; f; "/ D Œ.X; f0 ; 1/ ˝ Œ.; l/; where f0 W X ! RVn and l W ! are some definable functions 4.4 Extending Hrushovski–Loeser’s construction The morphisms hm and hQ m From now on, we shall denote by ŠK.RES/ the quotient of K.RES/ subject to the relations Œvalrv1 a/ D Œvalrv1 0/ for a in , and by ŠK.RES/ŒL loc the localization of ŠK.RES/ŒL with respect to the multiplicative family generated by Li , i Let m; n be in N, m 1, ; l/ in bdd Œn, and e in with me Z Set .m/ WD \ 1=mZ/n ; l;e WD l e/ Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM Lê, Regular and formal motivic Thom–Sebastiani theorem 11 and X ˛m ; l/ WD X e2;me2Z D X e2Z L m.j jCe/ L 1/n 2l;e m/ X L mj j e 1/n : L 2l;e=m m/ It is clear that ˛m ; l/ is an element of ŠK.RES/ŒL loc , and moreover, ˛m is independent of the choice of coordinates for n Indeed, let be the morphism in bdd from l;e ; ljl;e / to 0 ; l / Then j /j C l // D j j C l / D j j C e and the claim follows Thus ˛m defines a natural morphism of rings ˛m W K bdd / ! ŠK.RES/ŒL Q in vol bdd , one sets By using [14], for any X Q WD ˛Q m / L mj j L loc : 1/n Q 2.m/ and obtains a morphism of rings ˛Q m W K.vol bdd / ! ŠK.RES/ŒL Thus we can consider ˛m as an extension of ˛Q m ; moreover, X (4.2) ˛m ; l/ D ˛Q m l;e=m /L loc : e : loc e2Z We are able to construct a morphism ˇm W K RES/ ! ŠK.RES/ŒL by using Hrushovski–Loeser’s method Thanks to Remark 4.2, however, it suffices to define ˇm at elements of the form Œ.X; f; 1/ with X; f; 1/ an object in RES Assume that f X/ V V n , i.e., valrv fi x// D i for every x in X We set ´ ŒX .L Œ11 /mj j if m Zn ; ˇm X; f; 1/ WD otherwise: There are two steps to check that ker.˛Q m ˝ ˇm / contains ker.N0 /, where N0 is N reduced to the volume version (for the structure of K.volVFbdd /) These steps correspond to the factorization of N0 into K.vol bdd / ˝ K.volRES/ ! K.volRVbdd / and K.volVFbdd Œn/ ! K.volRVbdd Œn/=Œ11 ŒRV>0 1 : Hrushovski and Loeser [14] passed these by direct computation This can be applied to show that ker.˛m ˝ ˇm / contains ker.N / Consequently, from the tensor products ˛Q m ˝ ˇm and Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM 12 Lê, Regular and formal motivic Thom–Sebastiani theorem ˛m ˝ ˇm we obtain morphisms of rings hQ m W K.volVFbdd / ! ŠK.RES/ŒL hm W K VF bdd / ! ŠK.RES/ŒL loc ; loc : Moreover, there is a presentation of hm in terms of hQ m induced from (4.2) Namely, we have the following lemma whose proof is trivial and left to the reader Lemma 4.3 hm Œ.X; "// D Q P e=m//L e e2Z hm Œ" (in ŠK.RES/ŒL ) loc The morphism h We also use the morphisms from [14, Section 8.5] with their restriction, namely, ˛ W K.vol bdd / ! ŠK.RES/ŒL ; ˇ W K.volRES/ ! ŠK.RES/ŒL : By definition, ˇ.ŒX / D ŒX and ˛.Œ/ D /.L 1/n if is a definable subset of n , where is the o-minimal Euler characteristic in the sense of [13, Lemma 9.5] Since ker.˛˝ˇ/ contains ker.N0 / (cf [14]), it gives rise to a morphism of rings K.volVFbdd / ! ŠK.RES/ŒL : The composition of it with the localization morphism ŠK.RES/ŒL will be denoted by h 1 ! ŠK.RES/ŒL 1 loc P Proposition 4.4 The formal series Z X; "/.T / WD m hm Œ.X; "//T m is a rational function Moreover, we have limT !1 Z X; "/.T / D h.ŒX / Proof It is similar to the proof of [14, Proposition 8.5.1] 4.5 Endowing with a O -action and the morphisms hm , hQ m and h First, let us recall m D t , n [14, Section 4.3] Define a series ¹tm ºm by setting t1 D t , tnm For a k t //n definable set X over RES, we may assume X Vi1 =m Vin =m for some n, m and ij It is endowed with a natural action ı of m Now the k t 1=m //-definable function i1 in x1 ; : : : ; xn / 7! x1 =rv.tm /; : : : ; xn =rv.tm // maps X to a constructible subset Y of Ank , where Y is endowed with a ı The correspondence X 7! Y in its turn defines a morphism of rings ŠK.RES/ŒL ! ŠK0O Vark /ŒL m -action induced from (cf [13, Lemma 10.7], [14, Proposition 4.3.1]) Here, by definition, ŠK0O Vark / is the quotient of K0O Vark / by identifying all the classes ŒGm ; , where is a O -action on Gm induced by multiplication by roots of The previous morphism together with the natural one ŠK0O Vark /ŒL ! MkO Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM 13 Lê, Regular and formal motivic Thom–Sebastiani theorem induces the following morphisms of rings, both are denoted by ‚: ŠK.RES/ŒL ! MkO and ŠK.RES/ŒL O loc ! Mk;loc : Q m WD ‚ ı hQ m and h WD ‚ ı h with We now define ring morphisms hm WD ‚ ı hm , h O Q m and h starts from the same target Mk;loc In fact, while hm has the source K VFbdd /, h bdd K.volVF / Similarly to Lemma 4.3 and Proposition 4.4, we get Lemma 4.5 hm Œ.X; "// D P Q e2Z hm Œ" e=m//L e O ) (in Mk;loc P Proposition 4.6 The formal series Z.X; "/.T / WD m hm Œ.X; "//T m is a rational function Moreover, we have limT !1 Z.X; "/.T / D h.ŒX / 4.6 Description of the motivic Milnor fibers Regular case Let be in A definable subset X of VF` RV` is -invariant if, for 0 any x; x / VF` RV` and any y; y / VF` RV` with val.y/ , both x; x / and x; x / C y; y / simultaneously belong to either X or the complement of X in VF` RV` By [14, Lemma 3.1.1], any bounded definable subset of VF` that is closed in the valuation topology is -invariant for some in Assume that X is a -invariant definable subset of VFn RV` with 1=m/Z By [13], the set X.k t 1=m /// of k t 1=m //-points of X is the pullback of some definable subset XŒmI of kŒt 1=m =t /n RV` and the projection X ŒmI ! VFn is a finite-to-one map If is in with , the equality ŒX ŒmI D ŒX ŒmI Lnm / holds in ŠK.volRESŒn/, thus ŒX ŒmI L nm in ŠK.RES/ŒL is independent of the choice of large enough For brevity, we shall write XQ Œm for the quantity ŒX ŒmI L nm Cn as well as for its image under ‚ Proposition 4.7 (i) For X as previous, hQ m ŒX / D loc.XQ Œm/ (ii) Let f be a nonzero function on a d -dimensional smooth connected k-variety X, x a point of f 0/ Let be the reduction map X.R/ ! X.k/ Set ® ¯ X WD x X.R/ j x/ D x; rv.f x// D rv.t / : Then h.ŒX / D loc.Sf;x / (iii) For any in , h.Œ 1 / D and h.Œ 1 / D L (Note that Œ 1 and Œ 1 are the open and closed disks of valuative radius ) Proof (i) See Hrushovski–Loeser [14] (ii) We use [14, Corollary 8.4.2] for proving (ii) Since X is 2-invariant (it is in fact -invariant for any > in ), we have ® ¯ X ŒmI 2 D ' X.kŒt 1=m =.t // j '.0/ D x; rv.f '// D rv.t / : Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM 14 Lê, Regular and formal motivic Thom–Sebastiani theorem The condition rv.f '// D rv.t / is equivalent to f '/ Á t mod t mC1/=m , thus X ŒmI 2 is definably isomorphic via the map t 1=m 7! t to m 1/d1 ® ¯ ' X.kŒt =.t mC1 // j '.0/ D x; f '/ Á t m mod t mC1 Ak : We get hQ m ŒX / D loc.ŒX0;m L md1 / and the conclusion follows (iii) Assume D a=b with a; b in Z and a; b/ D Then ´ ma if n D mb; Q n Œa=b1 / D L h otherwise; thus h.Œa=b1 / D Also, ´ L hQ n Œa=b1 / D maC1 if n D mb; otherwise; thus h.Œa=b1 / D L Formal case Let X be a rigid k t //-variety which is the generic fiber of a special formal kŒŒt -scheme X, let ! be a gauge form on X We set b kŒŒt kŒŒt alg X WD X ˝ and X WD X ˝k t // k t //alg : The integer-valued function ordX !/ on X was already recalled in (3.2) Using the same way, one may define a rational-valued function ordX !/ on X , where ! is the pullback of ! via the natural morphism X ! X We denote this rational-valued function by val! Theorem 4.8 Let X be a relatively d -dimensional special formal kŒŒt -scheme with structural morphism f Let XÁ;rv (resp XÁ m/rv ) be a version of XÁ (resp XÁ m/) in which fÁ x/ D t is replaced by rvfÁ x/ D rv.t / (resp fÁ x/ Á t mod t mC1/=m ) Then, for any gauge form ! on XÁ , R (i) hm Œ.XÁ ; val! // D loc.Ld XÁ m/ j!.m/j/, R (ii) hm Œ.XÁ;rv ; val! // D loc.Ld XÁ m/rv j!.m/j/, (iii) h.ŒXÁ;rv / D h.ŒXÁ / As a consequence, for a closed point x of X0 and a gauge form ! on xŒ, R (iv) hm Œ.xŒ; val! // D loc.Ld xŒ j! m/j/, m R d (v) hm Œ.xŒrv ; val! // D loc.L xŒ j! m/j/, m;rv (vi) h.xŒrv / D h.xŒ/ D loc.Sf;x / Proof We prove (i) By Lemma 4.5, X Q m Œval! e=m//L h (4.3) hm Œ.XÁ ; val! // D e : e2Z Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM 15 Lê, Regular and formal motivic Thom–Sebastiani theorem By [14, Lemma 3.1.1], for each e in Z, there exists a invariant Thus it follows from Proposition 4.7 (i) that (4.4) e;m in such that val! e=m/ is e;m - E Q m Œval! e=m// D loc val! e=m/Œm h D loc val! e=m/ŒmI md L Cd for any Let Y ! X be a Néron smoothening for X, where Y is e;m in 1=m/Z a relatively d -dimensional kŒŒt -formal scheme topologically of finite type Then, XÁ D YÁ , since kŒŒt is henselian, so we can regard val! as a function on YÁ As val! is induced by the gauge form !, thus val! e=m/.m/ is a stable cylinder in Gr.Y.m// Moreover, ordY.m/ !.m// By definition of the measure (4.5) e/ D val! e=m/.m/: (cf (3.1)), we have ordY.m/ !.m// e/ D val! e=m/.m/ D val! e=m/ŒmI for 0 L in N large enough From (4.3), (4.4) and (4.5), it follows that  Z à  Z d hm Œ.XÁ ; val! // D loc L j!.m/j D loc Ld YÁ m/ XÁ m/ md à j!.m/j : This identity is also compatible with the canonical m -action by definition, thus it holds in O Mk;loc The identities (ii)–(vi) are direct consequences of the first one Remark 4.9 In [17], we define the motivic nearby cycles of a formal function f and denote it by Sf This is a virtual variety in the Grothendieck ring MXO of X0 -varieties with R good / is nothing but loc O -action In the context of Theorem 4.8 (iii), the quantity h.ŒX Á X0 Sf /, R O O where X0 is the forgetful (or pushforward) morphism MX0 ! Mk A new proof for the motivic Thom–Sebastiani theorem In this section, we give a model-theoretic proof for Theorem 2.1 by using the morphisms Q m and h For notational simplicity, we let f and g be regular functions on Ad1 and of rings h k Adk2 , vanishing at their origins, respectively Then, we shall prove that the following identity O holds in Mk;loc : (5.1) loc.Sf ˚g;.0;0/ / D loc Sf;0 Sg;0 C Sf;0 C Sg;0 : 5.1 Decomposition of the analytic Milnor fiber Consider the analytic Minor fiber of f ˚ g at the origin of Adk1 Adk2 : ® ¯ Z WD x; y/ md1 Cd2 j rv.f x/ C g.y// D rv.t / : Brought to you by | University of South Carolina Libraries Authenticated Download Date | 2/1/17 4:28 PM 16 Lê, Regular and formal motivic Thom–Sebastiani theorem This is a bounded 2-invariant definable subset of VFd1 Cd2 By Proposition 4.7 (ii), we have O h.ŒZ/ D loc.Sf ˚g;.0;0/ / in Mk;loc Let us decompose Z into a disjoint union of sets X, Y and Z subject to conditions valf x/ < valg.y/, valf x/ > valg.y/ and valf x/ D valg.y/, respectively In the sequel, we are going to compute h.ŒX /, h.ŒY /, h.ŒZ / and conclude Write ® ¯ X D x; y/ md1 Cd2 j rv.f x// D rv.t / as the product of the definable sets X WD ¹x md1 j rvf x/ D rv.t /º and md2 D Œ0d1 Statements (ii) and (iii) of Proposition 4.7 give h.ŒX1 / D loc.Sf;0 / and h.Œmd2 / D 1, thus O O Similarly, we also have h.ŒY / D loc.Sg;0 / in Mk;loc h.ŒX/ D loc.Sf;0 / in Mk;loc Set ® ¯ ® ¯ Z1 D x; y/ Z j valf x/ D ; Z