Splicing motivic zeta functions

Abstract

The first part of the paper discusses how to lift the splicing formula of Némethi and Veys to the motivic level. After defining the motivic zeta function and the monodromic motivic zeta function with respect to a differential form, we prove a splicing formula for them, which specializes to this formula of Némethi and Veys. We also show that we cannot introduce a monodromic motivic zeta functions in terms of a (splice) diagram since it does not contain all the necessary information. In the last part we discuss the generalized monodromy conjecture of Némethi and Veys. The statement also holds for motivic zeta functions but it turns out that the analogous statement for monodromic motivic zeta functions is not correct. We illustrate this with some examples.

Appendix: Existence of the Picard morphism

It turns out that the class of a smooth and proper algebraic variety in the Grothendieck ring of varieties determines the class of its Picard scheme. In this Appendix we will prove this statement using the argument described by Ekedahl [14]. We do this since [14] was never published and to clarify several steps in his argument.

Let X be a smooth and proper algebraic variety over $\mathbb {C}$. Recall that the Picard functor ${{\mathrm{Pic}}}_{X\slash \mathbb {C}}$ is a functor from the category of locally Noetherian $\mathbb {C}$-schemes to the category of abelian groups defined by the formula

$$\begin{aligned} {{\mathrm{Pic}}}_{X\slash \mathbb {C}}(T) := {{\mathrm{Pic}}}(X_T) \slash {{\mathrm{Pic}}}(T) \end{aligned}$$

where $X_T = X \times _\mathbb {C}T$. It turns out that the associated sheaf is representable by an abelian scheme whose identity component is an abelian variety and whose group of geometric components is finitely generated. This scheme will be denoted by ${{\mathrm{Pic}}}(X)$. See [15, Part 5] for more information on the Picard scheme.

Definition

Define $\mathcal {A}_{\mathbb {C}}$ as the abelian group generated by isomorphism classes of commutative algebraic group schemes over $\mathbb {C}$ whose identity component is an abelian variety and whose group of geometric components is finitely generated, subject to the relations $[A \oplus B ] = [ A] + [B]$, where A and B are such commutative algebraic group schemes over $\mathbb {C}$ and $A \oplus B$ is the direct product of A and B.

This leads us to the main result.

Theorem A.1

There is a (unique) group homomorphism ${{\mathrm{Pic}}}:K_0({{\mathrm{Var}}}_{{\mathbb {C}}}) \rightarrow \mathcal {A}_{\mathbb {C}}$ such that ${{\mathrm{Pic}}}( [X]) = [{{\mathrm{Pic}}}(X)]$ for every smooth proper variety X, and it extends to a morphism ${{\mathrm{Pic}}}:\hat{\mathcal {M}}_{\mathbb {C}} \rightarrow \mathcal {A}_{\mathbb {C}}$.

This theorem provides us with a new technique to compare elements in the Grothendieck ring. Using this theorem, we find the existence of $\widetilde{{{\mathrm{Pic}}}} :\hat{\mathcal {M}}_{\mathbb {C}} \rightarrow {A}_{\mathbb {C}}$ which sends a smooth complete variety to the class of its jacobian. It is obtained by composing ${{\mathrm{Pic}}}$ and the morphism $\mathcal {A}_{\mathbb {C}} \rightarrow {A}_{\mathbb {C}}$, where this last map is defined by sending the class of a commutative algebraic group scheme (whose identity component is an abelian variety) to the class of his identity component.

The keystone of the proof is Bittner’s presentation of $K_0({{\mathrm{Var}}}_{{\mathbb {C}}})$, which we restate here.

Theorem

[4, Theorem 3.1] The Grothendieck group of $\mathbb {C}$-varieties $K_0({{\mathrm{Var}}}_{{\mathbb {C}}})$ is isomorphic to the abelian group generated by the isomorphism classes of smooth projective $\mathbb {C}$-varieties subject to the relations $[\emptyset ] = 0$ and $[{{\mathrm{Bl}}}_YX] - [E] = [X] - [Y]$, where X is smooth and projective, $Y\subset X$ is a closed smooth subvariety, ${{\mathrm{Bl}}}_Y X$ is the blow-up of X along Y and E is the exceptional divisor of this blow-up.

This implies the following presentation of $\mathcal {M}_{\mathbb {C}} = K_0({{\mathrm{Var}}}_{{\mathbb {C}}})[\mathbb {L}^{-1}]$: $\mathcal {M}_{\mathbb {C}}$ is generated as a group by the elements $[X]\mathbb {L}^{-n}$ where X is smooth and projective and $n\in {\mathbb {Z}}_{>0}$, subject to the relations

$$\begin{aligned} \frac{ [X\times \mathbb {P}^{1}_{\mathbb {C}}] }{\mathbb {L}^{n+1} } = \frac{ [X] }{\mathbb {L}^{n} } + \frac{ [X ]}{\mathbb {L}^{n+1} }, \end{aligned}$$

where X is smooth and projective and $n\in {\mathbb {Z}}_{>0}$, and the relations

$$\begin{aligned} \frac{ [{{\mathrm{Bl}}}_YX]}{\mathbb {L}^n} - \frac{ [E]}{\mathbb {L}^n} = \frac{ [X]}{\mathbb {L}^n} - \frac{ [Y]}{\mathbb {L}^n}, \end{aligned}$$

where X, Y and E are as in the theorem and $n\in {\mathbb {Z}}_{>0}$.

Proof of Theorem A.1

We first show that the morphism defined by ${{\mathrm{Pic}}}( [X]) = [{{\mathrm{Pic}}}(X)]$ for X smooth and proper is well-defined as a map from $K_0({{\mathrm{Var}}}_{{\mathbb {C}}})$ to $\mathcal {A}_{\mathbb {C}}$. Using the Bittner representation, we need to show that

$$\begin{aligned} {{\mathrm{Pic}}}({{\mathrm{Bl}}}_YX) \oplus {{\mathrm{Pic}}}(Y) = {{\mathrm{Pic}}}(X) \oplus {{\mathrm{Pic}}}(E) \end{aligned}$$

where X, Y and E are as in the theorem. Hence we need to show this on the level of associated fppf sheaves. But a blow-up is preserved under base change by a flat morphism and thus ${{\mathrm{Pic}}}({{\mathrm{Bl}}}_{Y_T} X_T) = {{\mathrm{Pic}}}(X_T ) \oplus {\mathbb {Z}}$ and ${{\mathrm{Pic}}}(E_T) = {{\mathrm{Pic}}}( Y_T ) \oplus {\mathbb {Z}}$ for any flat morphism $T \rightarrow \mathbb {C}$ which induces the wanted isomorphism [17, Exercises 7.9 and 8.5 on pp. 170 and 188].

The next step is to define ${{\mathrm{Pic}}}' :\mathcal {M}_{\mathbb {C}} \rightarrow \mathcal {A}_{\mathbb {C}}$ and prove that this is actually an extension of ${{\mathrm{Pic}}}$. Define ${{\mathrm{Pic}}}'( [X] \mathbb {L}^{-n} )$ as

$$\begin{aligned} A_{X}^{0,n} \oplus A_{X}^{c,n} \end{aligned}$$

for a projective and smooth variety X and $n \in {\mathbb {Z}}_{>0}$, where $A_{X}^{c,n}$ is the inverse image of the classes of type $(n+1, n+1)$ under the map $H^{2n+2}(X,{\mathbb {Z}}) \rightarrow H^{2n+2}(X,\mathbb {C})$ and $A_{X}^{0,n}$ is the Weil intermediate Jacobian associated to the Hodge structure on $H^{2n+1}(X,{\mathbb {Z}})$. See [31] for more information. As discussed, we need to verify the two types of relations. Consider X to be a projective smooth $\mathbb {C}$-variety.

First we verify the blow-up relations. This is a consequence of [30, Theorem 7.31] and its proof since it induces that

$$\begin{aligned} H^{k}(X,{\mathbb {Z}}) \oplus \left( \bigoplus _{i=0}^{r-2} H^{k-2i - 2}(Y,{\mathbb {Z}}) \right) \oplus H^{k}(Y,{\mathbb {Z}}) \end{aligned}$$

is both isomorphic to $H^{k}({{\mathrm{Bl}}}_Y X,{\mathbb {Z}}) \oplus H^{k}(Y,{\mathbb {Z}})$ and $H^{k}(X ,{\mathbb {Z}}) \oplus H^{k}(E,{\mathbb {Z}})$ as Hodge structures, where r is the codimension of Y in X.

Remark that the cup-products map [30, Theorem 11.38] for X and $\mathbb {P}^{1}_{\mathbb {C}}$ induces an isomorphism

$$\begin{aligned} H^{i+2}(X \times \mathbb {P}^{1}_{\mathbb {C}} ,{\mathbb {Z}}) \cong H^{i+2}(X,{\mathbb {Z}}) \oplus H^{i}(X,{\mathbb {Z}}) \end{aligned}$$

as Hodge structures [12, p. 32].

Since the relations hold on the level of Hodge structures, they also hold for ${{\mathrm{Pic}}}'$ and thus ${{\mathrm{Pic}}}'$ is well-defined.

We show now that ${{\mathrm{Pic}}}'$ is indeed an extension of ${{\mathrm{Pic}}}$ and thus we will show that ${{\mathrm{Pic}}}(X) \cong A_{X}^{0,0} \oplus A_{X}^{c,0}$. Remark that the connected component ${{\mathrm{Pic}}}^0(X)$ is the classical Jacobian of X and thus isomorphic to $A_{X}^{0,0}$.

The Néron-Severi group ${{\mathrm{NS}}}(X)$ is defined by the short exact sequence

$$\begin{aligned} 0 \rightarrow {{\mathrm{Pic}}}^0(X) \rightarrow {{\mathrm{Pic}}}(X) \rightarrow {{\mathrm{NS}}}(X) \rightarrow 0 \end{aligned}$$

and is the group of components of ${{\mathrm{Pic}}}(X)$. This group can also be identified with the image of $d :H^{1}(X,\mathcal {O}_X^\times ) \rightarrow H^{2}(X,{\mathbb {Z}})$. Now Lefschetz’s theorem on (1, 1)-classes [16, p. 163] implies that ${{\mathrm{NS}}}(X)$ is exactly $A_{X}^{c,0}$.

Since we are working over an algebraically closed field and ${{\mathrm{NS}}}(X)$ is a discrete finitely generated abelian group, the short exact sequence splits and thus ${{\mathrm{Pic}}}(X) \cong {{\mathrm{Pic}}}^0(X) \oplus {{\mathrm{NS}}}(X)$.

To conclude we remark that $H^{n}(X,{\mathbb {Z}}) = 0$ if $n > 2 \hbox {dim}\,X$ and thus ${{\mathrm{Pic}}}'( [X] \mathbb {L}^{-n}) = 0$ if $\dim (X) -n \le 0$. This implies that ${{\mathrm{Pic}}}'$ sends every element of $F^0$ to 0 and thus ${{\mathrm{Pic}}}'$ can be extended to $\hat{\mathcal {M}}_{\mathbb {C}}$. $\square$

One of the results Ekedahl obtains with this is the fact that $\hat{\mathcal {M}}_{\mathbb {C}}$ is not a domain.