Birational Motivic Homotopy Theories and the Slice Filtration

TL;DR

This paper establishes an equivalence between the layers of the slice filtration in motivic homotopy theory and birational motivic categories, providing new insights into their structure and applications to projective spaces, Thom spaces, and blow ups.

Contribution

It introduces a novel approach to the slice filtration in the unstable motivic homotopy category, linking it with birational invariants and categories, and describes slices for key geometric objects.

Findings

Equivalence between slice filtration layers and birational motivic categories.

Descriptions of slices for projective spaces, Thom spaces, and blow ups.

Connection between the slice filtration and birational invariants.

Abstract

This paper is part of an endeavor to define an analogue of the slice filtration in the unstable motivic homotopy category. Our approach was inspired by the fact that the triangulated structures do not play a relevant role for the construction of birational homotopy categories as well as by the work of Kahn-Sujatha \cite{K-theory/0596} on birational motives, where the existence of a connection between the layers of the slice filtration and birational invariants is explicitly suggested. Our main result, shows that there is an equivalence of categories between the orthogonal components for the slice filtration and the birational motivic stable homotopy categories which are constructed in this paper. Relying on this equivalence, we are able to describe the slices for projective spaces (including $\mathbb P ^{\infty}$), Thom spaces and blow ups.