In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the no- tion of finite log correspondences, the dividing Nisnevich topology on log schemes, and the basic idea of parameterizing homotopies by □, i.e., the projective line with respect to its compactifying logarithmic structure at infinity. We show that Hodge cohomology of log schemes is a □-invariant theory that is representable in the category of loga- rithmic motives. Our category is closely related to Voevodsky’s category of motives and A1-invariant theories: assuming resolution of singularities, we identify the latter with the full subcategory comprised of A1-local objects in the category of logarithmic motives. Fundamental properties such as □-homotopy invariance, Mayer-Vietoris for coverings, the analogs of the Gysin sequence and the Thom space isomorphism as well as a blow-up formula and a projective bundle formula witness the robustness of the setup.

Triangulated categories of logarithmic motives over a field / F. Binda, D. Park, P.A. Oestvaer. - In: ASTÉRISQUE. - ISSN 0303-1179. - 433:(2022), pp. 1-280. [10.24033/ast.1172]

### Triangulated categories of logarithmic motives over a field

#### Abstract

In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the no- tion of finite log correspondences, the dividing Nisnevich topology on log schemes, and the basic idea of parameterizing homotopies by □, i.e., the projective line with respect to its compactifying logarithmic structure at infinity. We show that Hodge cohomology of log schemes is a □-invariant theory that is representable in the category of loga- rithmic motives. Our category is closely related to Voevodsky’s category of motives and A1-invariant theories: assuming resolution of singularities, we identify the latter with the full subcategory comprised of A1-local objects in the category of logarithmic motives. Fundamental properties such as □-homotopy invariance, Mayer-Vietoris for coverings, the analogs of the Gysin sequence and the Thom space isomorphism as well as a blow-up formula and a projective bundle formula witness the robustness of the setup.
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Ce texte a comme objectif la construction et l’étude d’une théorie des motifs mixtes pour les schémas logarithmiques sur un corps au sens de Fontaine, Illusie et Kato. Notre construction repose sur la notion de correspondance logarithmique finie, la topologie dividing-Nisnevich sur les schémas logarithmiques, et l’idée de paramétrer les homo- topies par □, c’est-à-dire la droite projective avec sa structure logarithmique standard à l’infini. On montre que la cohomologie de Hodge des schémas logarithmiques est une théorie □-invariante et représentable dans la catégorie des motifs logarithmiques. La catégorie des motifs mixtes de Voevodsky avec transferts se plonge de façon na- turelle (en supposant la résolution des singularités sur le corps de base) dans notre catégorie, et nous l’identifions avec sa sous-catégorie pleine donnée par les objects A1-locaux. Nous d ́montrons aussi des propriétés fondamentales comme des analogues du triangle de Gysin et d’éclatement, une formule du fibré projectif et un théorème d’isomorphisme de Thom.
Triangulated motives; logarithmic schemes; non A1-invariant cohomology theories; Hodge cohomology
Settore MAT/03 - Geometria
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2434/939927`