Burnside rings and volume forms with logarithmic poles

TL;DR

This paper develops a new theoretical framework for Burnside rings in algebraic geometry, focusing on varieties with logarithmic volume forms, introducing residue and specialization homomorphisms as invariants of birational morphisms.

Contribution

It introduces a novel theory of Burnside rings for algebraic varieties with logarithmic volume forms, including residue and specialization homomorphisms as new invariants.

Findings

Defined a residue homomorphism for varieties with logarithmic volume forms

Constructed an additive invariant of birational morphisms

Established a specialization homomorphism in this context

Abstract

We develop a theory of Burnside rings in the context of birational equivalences of algebraic varieties equipped with logarithmic volume forms. We introduce a residue homomorphism and construct an additive invariant of birational morphisms. We also define a specialization homomorphism. -- Nous proposons une th\'eorie d'anneaux de Burnside dans le contexte de la g\'eom\'etrie birationnelle des vari\'et\'es alg\'ebriques munies d'une forme volume \`a p\^oles logarithmiques. Nous introduisons un homomorphisme {\guillemotleft} r\'esidu {\guillemotright}, construisons un invariant additif des morphismes birationnels. Nous d\'efinissons aussi un homomorphisme de sp\'ecialisation.