Motivic zeta function

In algebraic geometry, the motivic zeta function of a smooth algebraic variety $X$ is the formal power series:

$Z(X,t)=\sum _{n=0}^{\infty }[X^{(n)}]t^{n}$

Here $X^{(n)}$ is the $n$-th symmetric power of $X$, i.e., the quotient of $X^{n}$ by the action of the symmetric group $S_{n}$, and $[X^{(n)}]$ is the class of $X^{(n)}$ in the ring of motives (see below).

If the ground field is finite, and one applies the counting measure to $Z(X,t)$, one obtains the local zeta function of $X$.

If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to $Z(X,t)$, one obtains $1/(1-t)^{\chi (X)}$.

Motivic measures

A motivic measure is a map $\mu$ from the set of finite type schemes over a field $k$ to a commutative ring $A$, satisfying the three properties

$\mu (X)\,$ depends only on the isomorphism class of $X$,

$\mu (X)=\mu (Z)+\mu (X\setminus Z)$ if $Z$ is a closed subscheme of $X$,

$\mu (X_{1}\times X_{2})=\mu (X_{1})\mu (X_{2})$.

For example if $k$ is a finite field and $A={\mathbb {Z} }$ is the ring of integers, then $\mu (X)=\#(X(k))$ defines a motivic measure, the counting measure.

If the ground field is the complex numbers, then Euler characteristic with compact supports defines a motivic measure with values in the integers.

The zeta function with respect to a motivic measure $\mu$ is the formal power series in $A[[t]]$ given by

$Z_{\mu }(X,t)=\sum _{n=0}^{\infty }\mu (X^{(n)})t^{n}$.

There is a universal motivic measure. It takes values in the K-ring of varieties, $A=K(V)$, which is the ring generated by the symbols $[X]$, for all varieties $X$, subject to the relations

$[X']=[X]\,$ if $X'$ and $X$ are isomorphic,

$[X]=[Z]+[X\setminus Z]$ if $Z$ is a closed subvariety of $X$,

$[X_{1}\times X_{2}]=[X_{1}]\cdot [X_{2}]$.

The universal motivic measure gives rise to the motivic zeta function.

Examples

Let $\mathbb {L} =[{\mathbb {A} }^{1}]$ denote the class of the affine line.

$Z({\mathbb {A} },t)={\frac {1}{1-{\mathbb {L} }t}}$

$Z({\mathbb {A} }^{n},t)={\frac {1}{1-{\mathbb {L} }^{n}t}}$

$Z({\mathbb {P} }^{n},t)=\prod _{i=0}^{n}{\frac {1}{1-{\mathbb {L} }^{i}t}}$

If $X$ is a smooth projective irreducible curve of genus $g$ admitting a line bundle of degree 1, and the motivic measure takes values in a field in which ${\mathbb {L} }$ is invertible, then

$Z(X,t)={\frac {P(t)}{(1-t)(1-{\mathbb {L} }t)}}\,,$

where $P(t)$ is a polynomial of degree $2g$. Thus, in this case, the motivic zeta function is rational. In higher dimension, the motivic zeta function is not always rational.

If $S$ is a smooth surface over an algebraically closed field of characteristic $0$, then the generating function for the motives of the Hilbert schemes of $S$ can be expressed in terms of the motivic zeta function by Göttsche's Formula

$\sum _{n=0}^{\infty }[S^{[n]}]t^{n}=\prod _{m=1}^{\infty }Z(S,{\mathbb {L} }^{m-1}t^{m})$

Here $S^{[n]}$ is the Hilbert scheme of length $n$ subschemes of $S$. For the affine plane this formula gives

$\sum _{n=0}^{\infty }[({\mathbb {A} }^{2})^{[n]}]t^{n}=\prod _{m=1}^{\infty }{\frac {1}{1-{\mathbb {L} }^{m+1}t^{m}}}$

This is essentially the partition function.