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On motivic zeta functions and the motivic nearby fiber. (English) Zbl 1085.14020
Let $$k$$ be an algebraically closed field of characteristic $$0$$ and $$X$$ a smooth connected variety of dimension $$d$$. Let $$f:X\to \mathbb A^1$$ be a function and $$S(f)(T)$$ the motivic zeta function defined by J. Denef and F. Loeser [J. Algebr. Geom. 7 (3), 505–537 (1998; Zbl 0943.14010)]. This zeta function is a formal power series with coefficients in a localized equivariant Grothendieck group of varieties over the zero locus of $$f$$. Let $$\mathbb L$$ be the class of the affine line in the Grothendieck group then the $$n$$-th coefficient of this series is given as $$\mathbb L^{-nd}$$ times the class of the variety of arcs $$\gamma(t)$$ of order $$n$$ on $$X$$ satisfying $$f\circ \gamma(t)=t^n$$. Some properties of the motivic zeta functions and the motivic nearby fibre are proved. Especially the relative dual of the motivic nearby fibre is computed.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 32S30 Deformations of complex singularities; vanishing cycles 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
##### Keywords:
motivic zeta function; motivic nearby fiber
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##### References:
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