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Valery Lunts (joint work with M. Larsen)

Motivic measures and stable birational geometry

A motivic Grothendieck group of $k$-varieties $K_0[\cal{V}_k]$ is generated by isomorphism classes of such varieties with relations $[X]=[Y]+[X-Y]$ for a closed $Y\subset X$. It is naturally a ring. A {\it motivic measure} is a ring homomorphism $\mu :K_0[\cal{V}_k]\to A$. Given a motivic measure $\mu$ Kapranov defines the zeta-function $Z_{\mu}(X,t)=\Sigma \mu (X^{(n)})t^n,$ where $X^(n)}$ is the $n$-th symmetric power of a $k$-variety $X$. This is a motivic analog of the classical Hasse-Weil zeta-function. Kapranov proves that $Z_{\mu}(X,t)$ is a rational function if $A$ is a field and $\dim X=1$. He also expects a similar result if $\dim X>1$. Recently Michael Larsen and I gave a negative solution to this problem. Our result is based on the study of $K_0[\cal{V}_{\bbC}]$ from the point of view of stable birational geometry.