Exponentiation of motivic measures
aa r X i v : . [ m a t h . AG ] D ec EXPONENTIATION OF MOTIVIC MEASURES
NIRANJAN RAMACHANDRAN AND GONC¸ ALO TABUADA
Abstract.
In this short note we establish some properties of all those motivicmeasures which can be exponentiated. As a first application, we show that therationality of Kapranov’s zeta function is stable under products. As a secondapplication, we give an elementary proof of a result of Totaro. Motivic measures
Let k be an arbitrary base field and Var( k ) the category of varieties , i.e. reducedseparated k -schemes of finite type. The Grothendieck ring of varieties K Var( k ) isdefined as the quotient of the free abelian group on the set of isomorphism classesof varieties [ X ] by the relations [ X ] = [ Y ] + [ X \ Y ], where Y is a closed subvarietyof X . The multiplication is induced by the product over Spec( k ). When k is ofpositive characteristic, one needs also to impose the relation [ X ] = [ Y ] for everysurjective radicial morphism X → Y ; see Mustat¸ˇa [19, Page 78]. Let L := [ A ].The structure of the Grothendieck ring of varieties is quite mysterious; see Poo-nen [21] for instance. In order to capture some of its flavor several motivic measures , i.e. ring homomorphisms µ : K Var( k ) → R , have been built. Examples include thecounting measure µ (see [19, Ex. 7.7]); the Euler characteristic measure χ c (see[19, Ex. 7.8]); the Hodge characteristic measure µ H (see [14, § µ P with values in Z [ u ] (see [14, § µ LL (see [13]); the Albanese measure µ Alb with values in thesemigroup ring of isogeny classes of abelian varieties (see [19, Thm. 7.21]); theGillet-Soul´e measure µ GS with values in the Grothendieck ring K (Chow( k ) Q ) ofChow motives (see [6]); and the measure µ NC with values in the Grothendieck ringof noncommutative Chow motives (see [23]). There exist several relations betweenthe above motivic measures. For example, χ c , µ H , µ P , µ NC , factor through µ GS .2. Kapranov’s zeta function
As explained in [19, Prop. 7.27], in the construction of the Grothendieck ringof varieties we can restrict ourselves to quasi-projective varieties. Given a motivicmeasure µ , Kapranov introduced in [11] the associated zeta function(2.0.1) ζ µ ( X ; t ) := ∞ X n =0 µ ([ S n ( X )]) t n ∈ (1 + R J t K ) × , where S n ( X ) stands for the n th symmetric product of the quasi-projective variety X . In the particular case of the counting measure, (2.0.1) agrees with the classicalWeil zeta function. Here are some other computations (with X smooth projective) ζ χ c ( X ; t ) = (1 − t ) − χ c ( X ) ζ P ( X ; t ) = Q r ≥ ( − u r t ) ( − br ζ Alb ( X ; t ) = [Alb( X )] t − t , where b r := dim C H rdR ( X ) and Alb( X ) is the Albanese variety of X ; see [22, § Date : July 17, 2018. Big Witt ring
Given a commutative ring R , recall from Bloch [2, Page 192] the construction ofthe big Witt ring W ( R ). As an additive group, W ( R ) is equal to ((1 + R J t K ) × , × ).Let us write + W for the addition in W ( R ) and 1 = 1 + 0 t + · · · for the zero element.The multiplication ∗ in W ( R ) is uniquely determined by the following requirements:(i) The equality (1 − at ) − ∗ (1 − bt ) − = (1 − abt ) − holds for every a, b ∈ R ;(ii) The assignment R W ( R ) is an endofunctor of commutative rings.The unit element is (1 − t ) − . We have also a (multiplicative) Teichm¨uller map R −→ W ( R ) a [ a ] := (1 − at ) − such that g ( t ) ∗ [ a ] = g ( at ) for every a ∈ R and g ( t ) ∈ W ( R ); see [2, Page 193]. Definition . Elements of the form p ( t ) − W q ( t ) ∈ W ( R ), with p ( t ) , q ( t ) ∈ R [ t ]and p (0) = q (0) = 1 ∈ R , are called rational functions .Let W rat ( R ) be the subset of rational elements. As proved by Naumann in [20,Prop. 6], W rat ( R ) is a subring of W ( R ). Moreover, R W rat ( R ) is an endofunctorof commutative rings. Recall also the construction of the commutative ring Λ( R ).As an additive group, Λ( R ) is equal to W ( R ). The multiplication is uniquelydetermined by the requirement that the involution group isomorphism ι : Λ( R ) → W ( R ) , g ( t ) g ( − t ) − , is a ring isomorphism. The unit element is 1 + t .4. Exponentiation
Let µ be a motivic measure. As explained by Mustat¸ˇa in [19, Prop. 7.28], theassignment X ζ µ ( X ; t ) gives rise to a group homomorphism(4.0.2) ζ µ ( − ; t ) : K Var( k ) −→ W ( R ) . Definition . ([22, § µ can be exponentiated if the abovegroup homomorphism (4.0.2) is a ring homomorphism. Corollary 4.2.
Given a motivic measure µ as in Definition 4.1, the following holds: (i) The ring homomorphism (4.0.2) is a new motivic measure; (ii)
Any motivic measure which factors through µ can also be exponentiated. This class of motivic measures is well-behaved with respect with rationality:
Proposition 4.3.
Let µ be a motivic measure as in Definition 4.1. If ζ µ ( X ; t ) and ζ µ ( Y ; t ) are rational functions, then ζ µ ( X × Y ; t ) is also a rational function.Proof. It follows automatically from the fact that W rat ( R ) is a subring of W ( R ). (cid:3) As proved by Naumann in [20, Prop. 8] (see also [22, Thm. 2.1]), the countingmeasure µ can be exponentiated. On the other hand, Larsen-Lunts “exotic”measure µ LL cannot be exponentiated! This would imply, in particular, that(4.0.3) ζ µ LL ( C × C ; t ) = ζ µ LL ( C ; t ) ∗ ζ µ LL ( C ; t )for any two smooth projective curves C and C . As proved by Kapranov in [11](see also [19, Thm. 7.33]), ζ µ ( C ; t ) is a rational function for every smooth projective Note that Kapranov’s zeta function is similar to the exponential function e x = P ∞ n =0 x n n ! .The product X n corresponds to x n and the symmetric product S n ( X ) corresponds to x n n ! since n ! is the size of the symmetric group on n letters. XPONENTIATION OF MOTIVIC MEASURES 3 curve C and motivic measure µ . Using Proposition 4.3, this hence implies that theright-hand side of (4.0.3) is also a rational function. On the other hand, as provedby Larsen-Lunts in [13, Thm. 7.6], the left-hand side of (4.0.3) is not a rationalfunction whenever C and C have positive genus. We hence obtain a contradiction.At this point, it is natural to ask which motivic measures can be exponen-tiated? We now provide a general answer to this question using the notion of λ -ring. Recall that a λ -ring R consists of a commutative ring equipped with asequence of maps λ n : A → A, n ≥
0, such that λ ( a ) = 1, λ ( a ) = a , and λ n ( a + b ) = P i + j = n λ i ( a ) λ j ( b ) for every a, b ∈ R . In other words, the map λ t : R −→ Λ( R ) a λ t ( a ) := X n λ n ( a ) t n is a group homomorphism. Equivalently, the composed map σ t : R λ t −→ Λ( R ) ι −→ W ( R ) a σ t ( a ) := λ − t ( a ) − (4.0.4)is a group homomorphism. This homomorphism is called the opposite λ -structure. Proposition 4.4.
Let µ be a motivic measure and R a λ -ring such that: (i) The above group homomorphism (4.0.4) is a ring homomorphism; (ii)
We have µ ([ S n ( X )]) = σ n ( µ ([ X ])) for every quasi-projective variety X .Under these conditions, the motivic measure µ can be exponentiated.Proof. Consider the following composed ring homomorphism(4.0.5) K Var( k ) µ −→ R σ t −→ W ( R ) . The equalities µ ([ S n ( X )]) = σ n ( µ ([ X ])) allow us to conclude that (4.0.5) agreeswith the group homomorphism ζ µ ( − ; t ). This achieves the proof. (cid:3) Remark . Let C be a Q -linear additive idempotent complete symmetric monoidalcategory. As proved by Heinloth in [9, Lem. 4.1], the exterior powers give rise to aspecial λ -structure on the Grothendieck ring K ( C ), with opposite λ -structure givenby the symmetric powers Sym n . In this case, (4.0.4) is a ring homomorphism. Remark . Let T ′ be a Q -linear thick triangulated monoidal subcategory ofcompact objects in the homotopy category T = Ho( C ) of a simplicial symmet-ric monoidal model category C . As proved by Guletskii in [8, Thm. 1], the exteriorpowers give rise to a special λ -structure on K ( T ′ ), with opposite λ -structure givenby the symmetric powers Sym n . In the case, (4.0.4) is a ring homomorphism. Remark . Assume that k is of characteristic zero. Thanks to Heinloth’s pre-sentation of the Grothendieck group of varieties (see [10, Thm. 3.1]), it suffices toverify the equality µ ([ S n ( X )]) = σ n ( µ ([ X ])) for every smooth projective variety X .As an application of the above Proposition 4.4, we obtain the following result: Proposition 4.8.
The Gillet-Soul´e motivic measure µ GS can be exponentiated.Proof. Recall from [6] that µ GS is induced by the symmetric monoidal functor(4.0.6) h : SmProj( k ) −→ Chow( k ) Q from the category of smooth projective varieties to the category of Chow motives.Since the latter category is Q -linear, additive, idempotent complete, and symmetricmonoidal, Remark 4.5 implies that the Grothendieck ring K (Chow( k ) Q ) satisfies NIRANJAN RAMACHANDRAN AND GONC¸ ALO TABUADA condition (i) of Proposition 4.4. As proved by del Ba˜no-Aznar in [4, Cor. 2.4], wehave h ( S n ( X )) ≃ Sym n h ( X ) for every smooth projective variety X . Using Remark4.7, this hence implies that condition (ii) of Proposition 4.4 is also satisfied. (cid:3) Remark . Thanks to Corollary 4.2(ii), all the motivic measures which factorthrough µ GS ( e.g. χ c , µ H , µ P , µ NC ) can also be exponentiated.5. Application I: rationality of zeta functions
By combining Propositions 4.3 and 4.8, we obtain the following result:
Corollary 5.1.
Let
X, Y be two varieties. If ζ µ GS ( X ; t ) and ζ µ GS ( Y ; t ) are rationalfunctions, then ζ µ GS ( X × Y ; t ) is also a rational function.Remark . Corollary 5.1 was independently obtained by Heinloth [9, Prop. 6.1] inthe particular case of smooth projective varieties and under the extra assumptionthat ζ µ GS ( X ; t ) and ζ µ GS ( Y ; t ) satisfy a certain functional equation. Example . Let
X, Y be smooth projective varieties ( e.g. abelian varieties) forwhich h ( X ) , h ( Y ) are Kimura-finite; see [12, § σ t : K (Chow( k ) Q ) −→ W ( K (Chow( k ) Q )) . As proved by Andr´e in [1, Prop. 4.6], σ t ([ h ( X )]) and σ t ([ h ( Y )]) are rational func-tions. Since ζ µ GS ( − ; t ) agrees with the composition of µ GS with (5.0.7), these latterfunctions are equal to ζ µ GS ( X ; t ) and ζ µ GS ( Y ; t ), respectively. Using Corollary 5.1,we hence conclude that ζ µ GS ( X × Y ; t ) is also a rational function.Recall from Voevodsky [24, § M c : Var( k ) p −→ DM gm ( k ) Q from the category of varieties and proper morphisms to the triangulated category ofgeometric motives. As proved in [24, Prop. 4.1.7], the functor (5.0.8) is symmetricmonoidal. Moreover, given a variety X and a closed subvariety Y ⊂ X , we have M c ( Y ) −→ M c ( X ) −→ M c ( X \ Y ) −→ M c ( Y )[1] ;see [24, Prop. 4.1.5]. Consequently, we obtain the following motivic measure K Var( k ) −→ K (DM gm ( k ) Q ) [ X ] [ M c ( X )] . (5.0.9) Proposition 5.4.
The above motivic measure (5.0.9) agrees with µ GS .Proof. As proved by Voevodsky in [24, Prop. 2.1.4], there exists a Q -linear additivefully-faithful symmetric monoidal functor(5.0.10) Chow( k ) Q −→ DM gm ( k ) Q such that (5.0.10) ◦ h ( X ) ≃ M c ( X ) for every smooth projective variety. Thanksto the work of Bondarko [3, Cor. 6.4.3 and Rk. 6.4.4], the above functor (5.0.10)induces a ring isomorphism K (Chow( k ) Q ) ≃ K (DM gm ( k ) Q ). Therefore, the prooffollows from Heinloth’s presentation of the Grothendieck ring of varieties in termsof smooth projective varieties; see [10, Thm. 3.1]. (cid:3) Thanks to Proposition 5.4, Example 5.3 admits the following generalization:
Example . Let
X, Y be varieties for which M c ( X ) , M c ( Y ) are Kimura-finite.Similarly to Example 5.3, ζ µ GS ( X × Y ; t ) is then a rational function. XPONENTIATION OF MOTIVIC MEASURES 5
In the above Examples 5.3 and 5.5, the rationality of ζ µ GS ( X × Y ; t ) can alter-natively be deduced from the stability of Kimura-finiteness under tensor products;see [12, § § Proposition 5.6.
Let X be a connected smooth projective surface over an alge-braically closed field k such that q = 0 and p g > , k := k ( X ) the functionfield of X , x a k -point of X , z the zero-cycle which is the pull-back of the cycle ∆( X ) − ( x × X ) along X × k → X × X , Z the support of z , and finally U thecomplement of Z in X = X × k . Under these notations, the following holds: (i) The geometric motive M c ( U ) is not Kimura-finite; (ii) Kapranov’s zeta function ζ µ GS ( U ; t ) is rational.Proof. As proved by O’Sullivan-Mazza in [18, Thm. 5.18], M ( U ) is not Kimura-finite. Since the surface U is smooth, we have M c ( U ) ≃ M ( U ) ∗ (2)[4]; see [24,Thm. 4.3.7]. Using the fact that − (2)[4] is an auto-equivalence of the categoryDM gm ( k ) Q and that M ( U ) ∗ is Kimura-finite if and only if M ( U ) is Kimura-finite(see Deligne [5, Prop. 1.18]), we conclude that M c ( U ) also is not Kimura-finite.We now prove item (ii). As proved by Guletskii in [8, § gm ( k ) Q satisfies the conditions of Remark 4.6. Consequently, we have a ring homomorphism(5.0.11) σ t : K (DM gm ( k ) Q ) −→ W ( K (DM gm ( k ) Q )) . As explained by Guletskii in [8, Ex. 5], σ t ([ M ( U )]) is a rational function. Thanks toLemma 5.7 below, we hence conclude that σ t ([ M c ( U )]) is also a rational function.The proof follows now from the fact that ζ µ GS ( − ; t ) agrees with the composition ofthe ring homomorphisms (5.0.9) and (5.0.11). (cid:3) Lemma 5.7.
Given a smooth variety X of dimension d , we have the equality σ t ([ M c ( X )]) = σ µ GS ( L ) d t ([ M ( X )]) . Proof.
The proof is given by the following identifications σ t ([ M c ( X )]) = σ t ([ M ( X ) ∗ ( d )[2 d ]])(5.0.12) = σ t ([ M ( X ) ∗ ] µ GS ( L d ))= σ t ([ M ( X ) ∗ ]) ∗ ζ µ GS ( L d ; t )= σ t ([ M ( X )]) ∗ ζ µ GS ( L d ; t )(5.0.13) = σ µ GS ( L ) d t ([ M ( X )]) , (5.0.14)where (5.0.12) follows from [24, Thm. 4.3.7], (5.0.13) from [5, Lem. 1.18], and(5.0.14) from Remark 6.2 below with µ := µ GS and g ( t ) := σ t ([ M ( X )]). (cid:3) Example . Let U , U be two surfaces as in Proposition 5.6. Thanks to the aboveCorollary 5.1, we hence conclude that ζ µ GS ( U × U ; t ) is a rational function. Remark . Thanks to Corollary 4.2(ii), the above Examples 5.3, 5.5, and 5.8,hold mutatis mutandis for any motivic measure which factors through µ GS .6. Application II: Totaro’s result
The following result plays a central role in the study of the zeta functions.
Proposition 6.1 (Totaro) . The equality ζ µ ( X × A n ; t ) = ζ µ ( X ; µ ( L ) n t ) holds forevery variety X and motivic measure µ . NIRANJAN RAMACHANDRAN AND GONC¸ ALO TABUADA
Its proof (see [7, Lem. 4.4][19, Prop. 7.32]) is non-trivial and based on a strati-fication of the symmetric products of X × A n . In all the cases where the motivicmeasure µ can be exponentiated, this result admits the following elementary proof: Proof.
Since [ X × A n ] = [ X ][ A n ] in the Grothendieck ring of varieties and themotivic measure µ can be exponentiated, the proof is given by the identifications ζ µ ( X × A n ; t ) = ζ µ ( X ; t ) ∗ ζ µ ( L n ; t )= ζ µ ( X ; t ) ∗ ζ µ ( L ; t ) ∗ n = ζ µ ( X ; t ) ∗ (1 + µ ( L ) t + µ ( L ) t + · · · ) ∗ n (6.0.15) = ζ µ ( X ; t ) ∗ ((1 − µ ( L ) t ) − ) ∗ n = ζ µ ( X ; t ) ∗ [ µ ( L )] ∗ n = ζ µ ( X ; t ) ∗ [ µ ( L ) n ]= ζ µ ( X ; µ ( L ) n t ) , where (6.0.15) follows from [19, Ex. 7.23] and [ µ ( L )] stands for the image of µ ( L ) ∈ R under the multiplicative Teichm¨uller map R → W ( R ). (cid:3) Remark . The above proof shows more generally that g ( t ) ∗ ζ µ ( L n ; t ) = g ( µ ( L ) n t )for every g ( t ) ∈ W ( R ) and motivic measure µ which can be exponentiated. Remark . (Fiber bundles) Given a fiber bundle E → X of rank n , we have[ E ] = [ X ][ A n ] in the Grothendieck ring of varieties; see [19, Prop. 7.4]. Therefore,the above proof, with X replaced by E , shows that ζ µ ( E ; t ) = ζ µ ( X ; µ ( L ) n t ). Remark . ( P n -bundles) Given a P n -bundle E → X , we have [ E ] = [ X ][ P n ] inthe Grothendieck ring of varieties; see [19, Ex. 7.5]. Therefore, by combining theequality [ P n ] = 1 + L + · · · + L n with the above proof, we conclude that ζ µ ( E ; t ) = ζ µ ( X ; t ) + W ζ µ ( X ; µ ( L ) t ) + W · · · + W ζ µ ( X ; µ ( L ) n t ) . G -varieties Let G be a finite group and Var G ( k ) the category of G -varieties , i.e. varieties X equipped with a G -action λ : G × X → X such that every orbit is contained in anaffine open set. The Grothendieck ring of G -varieties K Var G ( k ) is defined as thequotient of the free abelian group on the set of isomorphism classes of G -varieties[ X, λ ] by the relations [
X, λ ] = [
Y, τ ] + [ X \ Y, λ ], where (
Y, τ ) is a closed G -invariantsubvariety of ( X, λ ). The multiplication is induced by the product of varieties. Amotivic measure is a ring homomorphism µ G : K Var G ( k ) → R . As mentioned in[15, § χ c , µ H , µ P admit G -extensions χ Gc , µ G H , µ G P . Notation . Let Chow G ( k ) Q be the category of functors from the group G (con-sidered as a category with a single object) to the category Chow( k ) Q .Note that Chow G ( k ) Q is still a Q -linear additive idempotent complete symmetricmonoidal category and that (4.0.6) extends to a symmetric monoidal functor(7.0.16) h G : SmProj G ( k ) −→ Chow G ( k ) Q . Note also that the n th symmetric product of a G -variety is still a G -variety. There-fore, the notion of exponentiation makes sense in this generality. Gillet-Soul´e’smotivic measure µ GS admits the following G -extension: XPONENTIATION OF MOTIVIC MEASURES 7
Proposition 7.2.
The above functor (7.0.16) gives rise to a motivic measure µ G GS : K Var G ( k ) −→ K (Chow G ( k ) Q ) which can be exponentiated.Proof. Given a smooth projective variety X and a closed subvariety Y , let us denoteby Bl Y ( X ) the blow-up of X along Y and by E the associated exceptional divisor.As proved by Manin in [16, § h (Bl Y ( X )) ⊕ h ( Y ) ≃ h ( X ) ⊕ h ( E ) in Chow( k ) Q . Since this isomorphism is natural, it also holds inChow G ( k ) Q when X is replaced by a smooth projective G -variety ( X, λ ) and Y bya closed G -invariant subvariety ( Y, τ ). Therefore, thanks to Heinloth’s presentationof the Grothendieck ring of G -varieties in terms of smooth projective G -varieties(see [10, Lem. 7.1]), the assignment X h G ( X ) gives rise to a (unique) motivicmeasure µ G GS . The proof of Proposition 4.8, with (4.0.6) replaced by (7.0.16), showsthat this motivic measure µ G GS can be exponentiated. (cid:3) Remark . Thanks to Corollary 4.2(ii), all the motivic measures which factorthrough µ G GS ( e.g. χ Gc , µ G H , µ G P ) can also be exponentiated.Proposition 4.3 admits the following G -extension: Proposition 7.4.
Let µ G be a motivic measure which can be exponentiated and ( X, λ ) , ( Y, τ ) two G -varieties. If ζ µ G (( X, λ ); t ) and ζ µ G (( Y, τ ); t ) are rational func-tions, then ζ µ G (( X × Y, λ × τ ); t ) is also a rational function.Example . Assume that the group G (of order r ) is abelian and that the basefield k is algebraically closed of characteristic zero or of positive characteristic p with p ∤ r . Under these assumptions, Mazur proved in [17, Thm. 1.1] that ζ µ G (( C, λ ); t ) isa rational function for every smooth projective G -curve ( C, λ ) and motivic measure µ G . Thanks to Proposition 7.4, we hence conclude that ζ µ G (( C × C , λ × λ ); t ) isstill a rational function for every motivic measure µ G which can be exponentiatedand for any two smooth projective G -curves ( C , λ ) and ( C , λ ).Finally, Totaro’s result admits the following G -extension: Proposition 7.6.
Let µ G be a motivic measure which can be exponentiated and ( X.λ ) , ( A n , τ ) two G -varieties. When G (of order r ) is abelian and k is algebraicallyclosed, Kapranov’s zeta function ζ µ G (( X × A n , λ × τ ); t ) agrees with ζ µ G (cid:0) ( X, λ ); µ G ( S r ( A n , τ )) t (cid:1) + W ζ µ G (( X, λ ); t ) ∗ r − X l =0 n Y i =1 µ G ([ A , τ i ] · · · [ A , τ li ]) t l ! , where [ A n , τ ] = [ A , τ ] · · · [ A , τ n ] .Proof. Since [ X × A n , λ × τ ] = [ X, λ ][ A n , τ ] in the Grothendieck ring of G -varietiesand the motivic measure µ G can be exponentiated, we have the equality ζ µ G (( X × A n , λ × τ ); t ) = ζ µ G (( X, λ ); t ) ∗ ζ µ G (( A n , τ ); t ) . Moreover, as explained in [17, Page 1338], we have the following computation ζ µ G (( A n , τ ); t ) = 11 − µ G ( S r ( A n , τ )) t r − X l =0 n Y i =1 µ G ([ A , τ i ] · · · [ A , τ li ]) t l ! . Therefore, since (1 − µ G ( S r ( A n , τ )) t ) − is the Teichm¨uller class [ µ G ( S r ( A n , τ ))],the proof follows from the combination of the above equalities. (cid:3) NIRANJAN RAMACHANDRAN AND GONC¸ ALO TABUADA
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Niranjan Ramachandran, Department of Mathematics, University of Maryland, Col-lege Park, MD 20742 USA.
E-mail address : [email protected] URL : Gonc¸alo Tabuada, Department of Mathematics, MIT, Cambridge, MA 02139, USA
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