Motivic Gauss–Bonnet formulas

Levine, Marc; Raksit, Arpon

doi:10.2140/ant.2020.14.1801

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Motivic Gauss–Bonnet formulas

Marc Levine and Arpon Raksit

Vol. 14 (2020), No. 7, 1801–1851

DOI: 10.2140/ant.2020.14.1801

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 Déglise, Jin and Khan established a motivic Gauss–Bonnet formula relating this Euler characteristic to pushforwards of Euler classes in motivic cohomology theories. We apply this formula to $SL$ -oriented motivic cohomology theories to obtain explicit characterizations of this Euler characteristic. The main new input is a uniqueness result for pushforward maps in $SL$ -oriented theories, identifying these maps concretely in examples of interest.

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Keywords

motivic homotopy theory, Chow ring, Euler characteristics, hermitian K-theory

Mathematical Subject Classification 2010

Primary: 14F42

Secondary: 55N20, 55N35

Milestones

Received: 24 January 2019

Revised: 5 November 2019

Accepted: 23 February 2020

Published: 18 August 2020

Authors

Marc Levine
Fakultät Mathematik Universität Duisburg-Essen Essen Germany

Arpon Raksit
Department of Mathematics Stanford University Stanford, CA United States

Vol. 14, No. 7, 2020