Motivic and analytic nearby fibers at infinity and bifurcation sets

TL;DR

This paper employs motivic integration and non-archimedean geometry to analyze singularities at infinity of polynomial fibers, introducing motivic and analytic invariants that generalize classical bifurcation measures.

Contribution

It introduces a motivic invariant of fibers at infinity and a non-archimedean analytic nearby fiber, linking these to classical bifurcation sets and singularity measures.

Findings

Motivic nearby cycles at infinity generalize classical invariants.

Non-archimedean analytic nearby fibers recover motivic volumes.

Bifurcation sets from motivic and analytic invariants contain the classical bifurcation set.

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 $f\colon \mathbb A^d_\mathbb C \to \mathbb A^1_\mathbb C$. 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.