Motivic integration and Milnor fiber

Abstract

We put forward a uniform narrative that weaves together several variants of Hrushovski– Kazhdan style integral, and describe how it can facilitate the understanding of the Denef–Loeser motivic Milnor fiber and closely related objects. Our study focuses on the so-called “nonarchimedean Milnor fiber” that was introduced by Hrushovski and Loeser, and our thesis is that it is a richer embodiment of the underlying philosophy of the Milnor construction. The said narrative is first developed in the more natural complex environment, and is then extended to the real one via descent. In the process of doing so, we are able to provide more illuminating new proofs, free of resolution of singularities, of a few pivotal results in the literature, both complex and real. To begin with, the real motivic zeta function is shown to be rational, which yields the real motivic Milnor fiber; this is an analogue of the Hrushovski–Loeser construction. Then, applying $T$-convex integration after descent, matching the Euler characteristics of the topological Milnor fiber and the motivic Milnor fiber becomes a matter of simple computation, which is not only free of resolution of singularities as in the Hrushovski–Loeser proof, but is also free of other sophisticated algebro-geometric machineries. Finally, we also establish, in a much more intuitive manner, a new Thom–Sebastiani formula, which can be specialized to the one given by Guibert–Loeser–Merle.