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Motivic measures and stable birational geometry. (English) Zbl 1056.14015
Let $$K_0(\mathcal V_{\mathbb C})$$ be the Grothendieck ring of all algebraic varieties over $$\mathbb C$$. If $$X$$ is a complex variety and $$n\geq 1$$ is a natural number, denote by $$X^{(n)}$$ the $$n$$-fold symmetric of $$X$$. If $$\mu\colon K_0({\mathcal V}_{\mathbb C})\to A$$ is an $$A$$-valued motivic measure (i.e. a ring homomorphism) then one can define the formal power series $\zeta_{\mu}(X,t)=1+\sum_{n=1}^{\infty}\mu([X^{(n)}])t^n\in A[[t]],$ which is called the motivic zeta function of $$X$$. M. Kapranov [The elliptic curve in the $$S$$-duality theory and Eisenstein series for Kac-Moody groups, preprint, http://arxiv.org/math.AG/0001005] proved that if $$A$$ is a field and $$X$$ is a curve then $$\zeta_{\mu}(X,t)$$ is a rational function. Moreover he asked whether this is a rational function for all varieties $$X$$ of dimension $$\geq 2$$ . In the paper under review the authors give a negative answer to Kapranov’s question by proving the following.
Theorem: There is a field $$A$$ and a motivic measure $$\mu\colon K_0({\mathcal V}_{\mathbb C})\to A$$ such that for every smooth projective surface $$X$$ with $$p_g(X)=h^{2,0}(X)\geq 2$$, the motivic zeta function $$\zeta_{\mu}(X,t)$$ is not rational.

##### MSC:
 14E05 Rational and birational maps 14F42 Motivic cohomology; motivic homotopy theory
##### Keywords:
Grothendieck ring; motivic zeta function
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