Motivic Serre invariants, ramification, and the analytic Milnor fiber

Details Motivic Serre invariants, ramification, and the analytic Milnor fiber Motivic Serre invariants, ramification, and the analytic Milnor fiber

2007-03-08

arxiv.org/abs/math/0703217v2

Invent. Math. 168(1), 2007, 133-173

Mathematics

Algebraic Geometry

Some minor errors corrected. The original publication is available at http://www.springerlink.com

Scientific paper

10.1007/s00222-006-0029-7

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.

Affiliated with

Also associated with

No associations

Rating

Motivic Serre invariants, ramification, and the analytic Milnor fiber does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Motivic Serre invariants, ramification, and the analytic Milnor fiber, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Motivic Serre invariants, ramification, and the analytic Milnor fiber will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-642762