On Tate Milnor-Witt Motives

TL;DR

This paper studies Tate Milnor-Witt motives associated with smooth projective G_m-varieties, providing a splitting formula over Euclidean fields that simplifies their Chow-Witt group computations.

Contribution

It introduces a splitting formula for Tate Milnor-Witt motives over Euclidean fields, linking Chow-Witt groups to Chow groups and Witt sheaf cohomologies.

Findings

Tate Milnor-Witt motives exist for certain smooth projective G_m-varieties.

A splitting formula reduces Chow-Witt group calculations to more manageable invariants.

The approach applies to varieties with isolated rational fixed points.

Abstract

Smooth projective $\mathbb{G}_m$-varieties with isolated rational fixed points admit Tate Milnor-Witt motives. Over Euclidean fields, we give a splitting formula of such motives, which reduces the computation of their Chow-Witt groups to that of their Chow groups and cohomologies of Witt sheaf.