Motivic Gau{\ss}-Bonnet formulas

TL;DR

This paper explores motivic Gauß-Bonnet formulas within stable homotopy theory, providing explicit characterizations of Euler characteristics for smooth projective varieties using SL-oriented motivic cohomology theories.

Contribution

It introduces a unicity result for pushforward maps in SL-oriented theories, enabling concrete computations of Euler characteristics in motivic cohomology.

Findings

Explicit formulas for Euler characteristics in motivic cohomology

Unicity of pushforward maps in SL-oriented theories

Applications to smooth projective varieties

Abstract

The apparatus of motivic stable homotopy theory provides a notion of Euler characteristic for smooth projective varieties, valued in the Grothendieck-Witt ring of the base field. Previous work of the first author and recent work of D\'eglise-Jin-Khan establishes a "Gau\ss-Bonnet formula" relating this Euler characteristic to pushforwards of Euler classes in motivic cohomology theories. In this paper, we apply this formula to SL-oriented motivic cohomology theories to obtain explicit characterizations of this Euler characteristic. The main new input is a unicity result for pushforward maps in SL-oriented theories, identifying these maps concretely in examples of interest.