Motivic Gauß-Bonnet formulas

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-Khan establishes a "Gauß-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.

arXiv commentFinal version-to appear in Algebra & Number Theory

References in corpus (5)

• The six operations in equivariant motivic homotopy theory
• The homotopy coniveau tower
• A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula
• Algebraic K-theory, A^1-homotopy and Riemann-Roch theorems
• Push-forwards for Witt groups of schemes

Cited by in corpus (9)

• An arithmetic count of the lines meeting four lines in P^3
• Bézoutians and the ${\mathbb{A}}^1$-degree
• Conics meeting eight lines over perfect fields
• Power structures on the Grothendieck--Witt ring and the motivic Euler characteristic
• Trace maps in motivic homotopy and local terms
• The quadratic Euler characteristic of a smooth projective same-degree complete intersection
• Compactly supported ${\mathbb{A}}^1$-Euler characteristics of symmetric powers of cellular varieties
• Combing a hedgehog over a field
• The geometric diagonal of the special linear algebraic cobordism