In algebraic geometry, the motivic zeta function of a smooth algebraic variety X is the formal power series
Z ( X , t ) = ∑ n = 0 ∞ [ 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 ) χ ( X ) .
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
μ ( X ) depends only on the isomorphism class of
X ,
μ ( X ) = μ ( Z ) + μ ( X ∖ Z ) if
Z is a closed subscheme of
X ,
μ ( X 1 × X 2 ) = μ ( X 1 ) μ ( X 2 ) .
For example if k is a finite field and A = Z is the ring of integers, then μ ( 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 μ ( X , t ) = ∑ n = 0 ∞ μ ( 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 ∖ Z ] if
Z is a closed subvariety of
X ,
[ X 1 × X 2 ] = [ X 1 ] ⋅ [ X 2 ] .
The universal motivic measure gives rise to the motivic zeta function.
Let L = [ A 1 ] denote the class of the affine line.
Z ( A n , t ) = 1 1 − L n t Z ( P n , t ) = ∏ i = 0 n 1 1 − 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 L is invertible, then
Z ( X , t ) = P ( t ) ( 1 − t ) ( 1 − L t ) , where P ( t ) is a polynomial of degree 2 g . 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
∑ n = 0 ∞ [ S [ n ] ] t n = ∏ m = 1 ∞ Z ( S , 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
∑ n = 0 ∞ [ ( A 2 ) [ n ] ] t n = ∏ m = 1 ∞ 1 1 − L m + 1 t m This is essentially the partition function.
