We show how formal and rigid geometry can be used in the theory of complex
singularities, and in particular in the study of the Milnor fibration and the
motivic zeta function. We introduce the so-called analytic Milnor fiber
associated to the germ of a morphism f from a smooth complex algebraic variety
X to the affine line. This analytic Milnor fiber is a smooth rigid variety over
the field of Laurent series C((t)). Its etale cohomology coincides with the
singular cohomology of the classical topological Milnor fiber of f; the
monodromy transformation is given by the Galois action. Moreover, the points on
the analytic Milnor fiber are closely related to the motivic zeta function of
f, and the arc space of X.
We show how the motivic zeta function can be recovered as some kind of Weil
zeta function of the formal completion of X along the special fiber of f, and
we establish a corresponding Grothendieck trace formula, which relates, in
particular, the rational points on the analytic Milnor fiber over finite
extensions of C((t)), to the Galois action on its etale cohomology.
The general observation is that the arithmetic properties of the analytic
Milnor fiber reflect the structure of the singularity of the germ f.Comment: Some minor errors corrected. The original publication is available at
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Nous généralisons la théorie de l'intégration motivique au cadre des schémas formels. Nous définissons etétudions l'anneau booléen des ensembles mesurables, la mesure motivique, l'intégrale motivique et nous démontrons un théorème de changement de variables pour cette intégrale. Abstract (Motivic Integration on Formal Schemes).-We generalize the theory of motivic integration on formal schemes. In particular, we define and study the boolean ring of mesurable subsets, the motivic measure, the motivic integral and we prove a theorem of change of variables for this integral.
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