Vol. 14, No. 7, 2020

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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.

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