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 Xⁿ 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)^χ(X).

Motivic measures

A motivic measure is a map μ 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 $A={\Bbb Z}$ 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 μ 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 $\Bbb L=[{\Bbb A}^1]$ denote the class of the affine line.

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

$Z({\Bbb P}^n,t)=\prod_{i=0}^n\frac{1}{1-{\Bbb L}^i t}$

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 ${\Bbb L}$ is invertible, then

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

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

$\sum_{n=0}^\infty[S^{[n]}]t^n=\prod_{m=1}^\infty Z(S,{\Bbb L}^{m-1}t^m)$

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

$\sum_{n=0}^\infty[({\Bbb A}^2)^{[n]}]t^n=\prod_{m=1}^\infty \frac{1}{1-{\Bbb L}^{m+1}t^m}$

This is essentially the partition function.