Nearby cycles at infinity as a triangulated functor

Abstract

For a polynomial function $f \colon \mathbb{C}^n \longrightarrow \mathbb{C}$, it is well-known in singularity theory (after Thom, Pham, Verdier,...) that outside a finite subset of $\mathbb{C}$, the function is a locally trivial $C^{\infty}$-fibration. The minimal such finite set is called the bifurcation set associated with $f$ and determining the bifurcation sets is a difficult task in singularity theory. To such a function, Raibaut attaches a virtual invariant called motivic nearby cycles at infinity. This invariant lives in some Grothendieck ring of varieties and measures the difference between the Euler characteristics of the general fiber and a fixed fiber. In this work, we show that the motivic nearby cycles at infinity admits a functorial version in the context of motivic homotopy theory, called the motivic nearby cycles functors at infinity. The motivic nearby cycles functors at infinity live in the world of motives and hence capture cohomological information (not just Euler characteristics) of singularities at infinity and realizes to Raibaut's construction in the world of virtual motives. Moreover, our construction is universal in the sense that it is applicable to any theory of nearby cycles functors defined in terms of six operations.