# K\"unneth formulas for motives and additivity of traces

## TL;DR

This paper establishes K"unneth formulas and additivity properties for motives in motivic homotopy categories, leading to a characteristic class invariant related to Euler characteristics, with applications to relative settings.

## Contribution

It proves new K"unneth formulas and additivity of the Verdier pairing in motivic categories, and characterizes the characteristic class of motives under a Chow weight structure.

## Key findings

- Proved K"unneth formulas in motivic homotopy categories.
- Established additivity of the Verdier pairing using derivators.
- Defined the relative characteristic class under transversality conditions.

## Abstract

We prove several K\"unneth formulas in motivic homotopy categories and deduce a Verdier pairing in these categories following SGA5, which leads to the characteristic class of a constructible motive, an invariant closely related to the Euler-Poincar\'e characteristic. We prove an additivity property of the Verdier pairing using the language of derivators, following the approach of May and Groth-Ponto-Shulman; using such a result we show that in the presence of a Chow weight structure, the characteristic class for all constructible motives is uniquely characterized by proper covariance, additivity along distinguished triangles, refined Gysin morphisms and Euler classes. In the relative setting, we prove the relative K\"unneth formulas under some transversality conditions, and define the relative characteristic class.