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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)
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