Motivic infinite cyclic covers

TL;DR

This paper introduces a new motivic invariant called the motivic infinite cyclic cover, associated with certain infinite cyclic covers of complex manifolds, unifying various motivic Milnor fiber concepts and proving its birational invariance.

Contribution

It constructs the motivic infinite cyclic cover in the Grothendieck ring and demonstrates its birational invariance, unifying different motivic Milnor fiber notions.

Findings

Provides a new motivic invariant for infinite cyclic covers

Unifies the motivic Milnor fiber of hypersurface singularities and rational functions

Establishes birational invariance of the motivic infinite cyclic cover

Abstract

We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (assuming certain finiteness conditions are satisfied) an element in the Grothendieck ring $K_0({\rm Var}^{\hat \mu}_{\mathbb{C}})$, which we call {\it motivic infinite cyclic cover}, and show its birational invariance. Our construction provides a unifying approach for the Denef-Loeser motivic Milnor fibre of a complex hypersurface singularity germ, and the motivic Milnor fiber of a rational function, respectively.