Motivic zeta function

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

Here is the -th symmetric power of , i.e., the quotient of by the action of the symmetric group , and is the class of in the ring of motives (see below).

If the ground field is finite, and one applies the counting measure to , one obtains the local zeta function of .

If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to , one obtains .

Motivic measures[edit]

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

depends only on the isomorphism class of ,
if is a closed subscheme of ,
.

For example if is a finite field and is the ring of integers, then 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 is the formal power series in given by

.

There is a universal motivic measure. It takes values in the K-ring of varieties, , which is the ring generated by the symbols , for all varieties , subject to the relations

if and are isomorphic,
if is a closed subvariety of ,
.

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

Examples[edit]

Let denote the class of the affine line.

If is a smooth projective irreducible curve of genus admitting a line bundle of degree 1, and the motivic measure takes values in a field in which is invertible, then

where is a polynomial of degree . Thus, in this case, the motivic zeta function is rational. In higher dimension, the motivic zeta function is not always rational.

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

Here is the Hilbert scheme of length subschemes of . For the affine plane this formula gives

This is essentially the partition function.