Motivic zeta function

In algebraic geometry, the motivic zeta function of a smooth algebraic variety [math]\displaystyle{ X }[/math] is the formal power series

[math]\displaystyle{ Z(X,t)=\sum_{n=0}^\infty [X^{(n)}]t^n }[/math]

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

If the ground field is finite, and one applies the counting measure to [math]\displaystyle{ Z(X,t) }[/math], one obtains the local zeta function of [math]\displaystyle{ X }[/math].

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

Motivic measures

A motivic measure is a map [math]\displaystyle{ \mu }[/math] from the set of finite type schemes over a field [math]\displaystyle{ k }[/math] to a commutative ring [math]\displaystyle{ A }[/math], satisfying the three properties

[math]\displaystyle{ \mu(X)\, }[/math] depends only on the isomorphism class of [math]\displaystyle{ X }[/math],

[math]\displaystyle{ \mu(X)=\mu(Z)+\mu(X\setminus Z) }[/math] if [math]\displaystyle{ Z }[/math] is a closed subscheme of [math]\displaystyle{ X }[/math],

[math]\displaystyle{ \mu(X_1\times X_2)=\mu(X_1)\mu(X_2) }[/math].

For example if [math]\displaystyle{ k }[/math] is a finite field and [math]\displaystyle{ A={\mathbb Z} }[/math] is the ring of integers, then [math]\displaystyle{ \mu(X)=\#(X(k)) }[/math] 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 [math]\displaystyle{ \mu }[/math] is the formal power series in [math]\displaystyle{ At }[/math] given by

[math]\displaystyle{ Z_\mu(X,t)=\sum_{n=0}^\infty\mu(X^{(n)})t^n }[/math].

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

[math]\displaystyle{ [X']=[X]\, }[/math] if [math]\displaystyle{ X' }[/math] and [math]\displaystyle{ X }[/math] are isomorphic,

[math]\displaystyle{ [X]=[Z]+[X\setminus Z] }[/math] if [math]\displaystyle{ Z }[/math] is a closed subvariety of [math]\displaystyle{ X }[/math],

[math]\displaystyle{ [X_1\times X_2]=[X_1]\cdot[X_2] }[/math].

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

Examples

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

[math]\displaystyle{ Z({\mathbb A},t)=\frac{1}{1-{\mathbb L} t} }[/math]

[math]\displaystyle{ Z({\mathbb A}^n,t)=\frac{1}{1-{\mathbb L}^n t} }[/math]

[math]\displaystyle{ Z({\mathbb P}^n,t)=\prod_{i=0}^n\frac{1}{1-{\mathbb L}^i t} }[/math]

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

[math]\displaystyle{ Z(X,t)=\frac{P(t)}{(1-t)(1-{\mathbb L}t)}\,, }[/math]

where [math]\displaystyle{ P(t) }[/math] is a polynomial of degree [math]\displaystyle{ 2g }[/math]. Thus, in this case, the motivic zeta function is rational. In higher dimension, the motivic zeta function is not always rational.

If [math]\displaystyle{ S }[/math] is a smooth surface over an algebraically closed field of characteristic [math]\displaystyle{ 0 }[/math], then the generating function for the motives of the Hilbert schemes of [math]\displaystyle{ S }[/math] can be expressed in terms of the motivic zeta function by Göttsche's Formula

[math]\displaystyle{ \sum_{n=0}^\infty[S^{[n]}]t^n=\prod_{m=1}^\infty Z(S,{\mathbb L}^{m-1}t^m) }[/math]

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

[math]\displaystyle{ \sum_{n=0}^\infty[({\mathbb A}^2)^{[n]}]t^n=\prod_{m=1}^\infty \frac{1}{1-{\mathbb L}^{m+1}t^m} }[/math]

This is essentially the partition function.