Motivic and analytic nearby fibers at infinity and bifurcation sets

Abstract

In this paper we use motivic integration and non-archimedean analytic geometry to study the singularities at infinity of the fibers of a polynomial map $f:{\mathbb{A}}_{\mathbb{C}}^d\to {\mathbb{A}}_{\mathbb{C}}^1$. We show that the motive $S_{f,a}^{\infty }$ of the motivic nearby cycles at infinity of $f$ for a value $a$ is a motivic generalization of the classical invariant ${\lambda }_f(a)$, an integer that measures a lack of equisingularity at infinity in the fiber $f^{-1}(a)$. We then introduce a non-archimedean analytic nearby fiber at infinity ${\mathcal{F}}_{f,a}^{\infty }$ whose motivic volume recovers the motive $S_{f,a}^{\infty }$. With each of $S_{f,a}^{\infty }$ and ${\mathcal{F}}_{f,a}^{\infty }$ can be naturally associated a bifurcation set; we show that the first one always contains the second one, and that both contain the classical topological bifurcation set of $f$ whenever $f$ has isolated singularities at infinity.

18 pages