Let $X$ be a smooth projective variety over $\mathbb{F}_{q}$. For each pair of positive integers $n$ and $d$, let $\text{Chow}_{n,d}(X)$ denote the (coarse) moduli space of $n$-cycles of degree $d$ on $X$, as defined in §3 of Kollár's book Rational Curves on Algebraic Varieties. Fixing $n>0$, define $Z_{n}(X,t)$ to be the formal power series $$Z_{n}(X,t)\ \overset{{}_{\text{def}}}{=}\ \sum_{d=0}^{\infty}\big|\text{Chow}_{n,d}(X)(\mathbb{F}_{q})\big|\ \!t^{d}.$$ What is known about this series? Does it have obvious pathological behavior? Are there natural candidates for analogues of the Weil conjectures that one expects for $Z_{n}(X,t)$? Is there an easy way to derive any of these analogues from the Weil conjectures themselves, i.e., from the $n=0$ case?
1 Answer
As Jason Starr notes in his comment, the series I ask about cannot be rational in general. There's more to the story though. I recently stumbled across several papers where Elizondo and Kimura study motivic versions of these series:
[1] Elizondo, E. Javier. The Euler Series of Restricted Chow Varieties.
[2] Elizondo, E. Javier and Kimura, Shun-Ichi. Irrationality of motivic series of Chow varieties.
[3] Elizondo, E. Javier and Kimura, Shun-Ichi. Rationality of motivic Chow series modulo $\mathbb{A}^{\!1}$-homotopy.
I share some of their results here.
Fix a ground field $k$. Let $\mathcal{M}^{\text{rat}}_{k}$ denote the category of Chow motives, and let $K_{0}(\mathcal{M}^{\text{rat}}_{k})$ denote its Grothendieck ring. Let $K(\text{Var}_{k})$ denote the Grothendieck ring of varieties. If $\overline{k}=k$ and $\text{char}_{\ \!}k=0$,then we have a ring homomorphism $K(\text{Var}_{k})\longrightarrow K_{0}(\mathcal{M}^{\text{rat}}_{k})$ taking the class of each smooth projective $k$-variety to the class of its Chow motive. In each of the rings of formal power series $K_{0}(\mathcal{M}^{\text{rat}}_{k})[\![t]\!]$ or $K(\text{Var}_{k})[\![t]\!]$, we can consider a motivic version of the series I ask about, namely
$$
\sum_{d=0}^{\infty}\text{Chow}_{n,d}(X)\ t^{d},
$$
where the coefficients $\text{Chow}_{n,d}(X)$ denote either the $K_{0}(\mathcal{M}^{\text{rat}}_{k})$- or $K(\text{Var}_{k})$-classes of the corrresponding Chow varieities. In [2, Theorem 4.5], Elizondo and Kimura show that if $n\ge 2$, then the motivic Chow series
$$
\sum_{d=0}^{\infty}\text{Chow}_{n-1,d}(\mathbb{P}^{n})\ \!t^{d}\ =\ \sum_{d=0}^{\infty}\frac{1-{\mathbb{A}^{\!1}}^{\left(\!\begin{smallmatrix} n+d \\ d\end{smallmatrix}\!\right)}}{1-\mathbb{A}^{1}}\ \!t^{d}
$$
in $K_{0}(\mathcal{M}^{\text{rat}}_{k})[\![t]\!]$ is irrational. But then in [3], they make the following observation: If we imagine giving $\mathbb{A}^{1}$ some family of measures $\mu(\mathbb{A}^{1})$ such that the limit $\mu(\mathbb{A}^{1})\to 1$ makes sense, then in this limit the above series becomes
$$
\frac{1}{(1-t)^{n+1}}.
$$
In particular, it is rational. This leads to the idea of passing to an "$(\mathbb{A}^{\!1}\to 1)$-limit" , which we can think of as being some kind of motivic version of a deformation retraction of $\mathbb{A}^{1}$ to a point. Concretely, this means passing to one of the quotients $K_{0}(\mathcal{M}^{\text{rat}}_{k})_{\mathbb{A}^{1}}\overset{{}_{\text{def}}}{=}K_{0}(\mathcal{M}^{\text{rat}}_{k})\big/\!\sim$ or $K(\text{Var}_{k})_{\mathbb{A}^{1}}\overset{{}_{\text{def}}}{=}K(\text{Var}_{k})\big/\!\sim$, where "$\sim$" is the relation
$$
X\times\mathbb{A}^{1}\ \sim\ X.
$$
Assume $\overline{k}=k$ and $\text{char}_{\ \!}k=0$. If $X$ is a smooth projective $k$-variety, then for each $n\in \mathbb{Z}_{\ge0}$, let $B_{n}(X)$ denote the commutative monoid of connected components of the scheme $\text{Chow}_{n}(X)\overset{{}_{\text{def}}}{=}\bigsqcup_{d=0}^{\infty}\text{Chow}_{n,d}(X)$. For each $\beta\in B_{n}(X)$, let $\text{Chow}_{n,\beta}(X)$ denote the corresponding component of $\text{Chow}_{n}(X)$. Then it is somewhat more natural to consider the motivic Chow series
$$
\sum_{\beta\in B_{n}(X)}^{\infty}\!\!\!\!\text{Chow}_{n,\beta}(X)\ t^{\beta}
$$
inside either of the rings
$$
K(\text{Var}_{k})_{\mathbb{A}^{1}}\big[\!\big[B_{n}(X)\big]\!\big]
\ \ \ \ \ \ \mbox{or}\ \ \ \ \ \
K_{0}(\mathcal{M}^{\text{rat}}_{k})_{\mathbb{A}^{1}}\big[\!\big[B_{n}(X)\big]\!\big]
$$
Elizondo and Kimura prove the following:
If $C_{1}$ and $C_{2}$ are very general curves, then for each $n\ge 0$, the motivic Chow series $$ \sum_{\beta\in B_{n}(C_{1}\times C_{2})}^{\infty}\!\!\!\!\!\!\!\!\!\text{Chow}_{n,\beta}(C_{1}\times C_{2})\ t^{\beta} $$ is rational [3, Corollary 3.11].
If $Y_{\Sigma}$ is a smooth projective toric variety, then for each $n\in\mathbb{Z}_{\ge0}$, its motivic Chow series in $K_{0}(\mathcal{M}^{\text{rat}}_{k})_{\mathbb{A}^{1}}\big[\!\big[B_{n}(Y_{\Sigma})\big]\!\big]$ is rational [3, Corollary 4.4], and satisfies the formula $$ \sum_{\beta\in B_{n}(Y_{\Sigma})}^{\infty}\!\!\!\!\!\!\text{Chow}_{n,\beta}(Y_{\Sigma})\ t^{\beta} \ \ = \prod_{\sigma\in\Sigma(n)}\frac{1}{1-t^{\beta(\sigma)}}, $$ where $\beta(\sigma)$ denotes the $B_{n}(Y_{\Sigma})$-class of the torus-orbit corresponding to $\sigma$. Because the Euler characteristic preserves $\mathbb{A}^{\!1}$-homotopy, this provides a motivic argument behind Elizondo's earlier proof that each series $$ \sum_{d=0}^{\infty}\chi\big(\text{Chow}_{n,d}(Y_{\Sigma})\big)\ \!t^{d} \ \ \in\ \ \mathbb{Z}[\![t]\!] $$ is rational [1]. It also generalizes the formula for the motivic L-function, modulo $\mathbb{A}^{\!1}$-homotopy, of a smooth projective toric variety $Y_{\Sigma}$ that comes from the motivic decomposition of $Y_{\Sigma}$.