Parametrized Kähler class and Zariski dense orbital 1-cohomology

Let $\Gamma$ be a finitely generated group and let $(X,\mu_X)$ be an ergodic standard Borel probability $\Gamma$-space. Suppose that $\mathcal{X}$ is a Hermitian symmetric space not of tube type and assume that $G=\operatorname{Isom}(\mathcal{X})^{\circ}$ is simple. Given a Zariski dense measurable cocycle $\sigma:\Gamma \times X \to G$, we define the notion of parametrized Kähler class and we show that it completely determines the cocycle up to cohomology.

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The first author is funded by the European Union - NextGenerationEU under the National Recovery and Resilience Plan (PNRR) - Mission 4 Education and research - Component 2 From research to business - Investment 1.1 Notice Prin 2022 - DD N. 104 del 2/2/2022, from title “Geometry and topology of manifolds”, proposal code 2022NMPLT8 - CUP J53D23003820001.

Received 10 June 2021

Received revised 2 September 2022

Accepted 23 October 2022

Published 17 July 2024