Orbital cohomology and Kähler rigidity

Alessio Savini

doi:10.5802/tsg.384

Alessio Savini ¹

¹ Section de Mathématiques, University of Geneva, Rue Du Conseil Géneral 7-9, Geneva 1205 (Switzerland)

Séminaire de théorie spectrale et géométrie, Tome 37 (2021-2022), pp. 111-135.

Résumé

In the late $70$ ’s Feldman and Moore [7] defined the cohomology associated to a countable equivalence relation with coefficients in an Abelian Polish group. When the equivalence relation is the orbital one, that is it is induced by a measure preserving action of a countable group $Γ$ on a standard Borel probability space $(X, μ)$ , it still makes sense to consider the Feldmann–Moore $1$ -cohomology with $G$ -coefficients, where this time $G$ can be any topological group. The latter cohomology, denoted by $H^{1} (Γ ↷ X; G)$ , is very misterious and hard to compute, except for some exceptional cases.

In this expository paper we are going to focus our attention on the particular case when $Γ$ is a finitely generated group and $G$ is a Hermitian Lie group. We are going to give some recent rigidity results in this context and we will see how those results can be used to say something relevant about (some subsets of) the orbital cohomology.

Publié le : 2025-07-23

DOI : 10.5802/tsg.384

Affiliations des auteurs :

Alessio Savini ¹

¹ Section de Mathématiques, University of Geneva, Rue Du Conseil Géneral 7-9, Geneva 1205 (Switzerland)

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Alessio Savini. Orbital cohomology and Kähler rigidity. Séminaire de théorie spectrale et géométrie, Tome 37 (2021-2022), pp. 111-135. doi : 10.5802/tsg.384. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.384/

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