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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.

14E05 Rational and birational maps
14F42 Motivic cohomology; motivic homotopy theory
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