Cousin groups and Hodge structures

TL;DR

This paper explores the geometric properties of Hodge Cousin groups, especially those linked to polarized rational Hodge structures, and introduces abelian Cousin groups with a Poincaré reducibility analogue.

Contribution

It introduces the class of abelian Cousin groups and proves an analogue of Poincaré complete reducibility for them.

Findings

Characterization of Hodge Cousin groups related to polarized Q-Hodge structures

Introduction of abelian Cousin groups

Proof of Poincaré complete reducibility analogue

Abstract

We consider the geometric properties of Hodge Cousin groups, introduced in an unpublished paper \cite{OVV}, emphasizing the case of Hodge Cousin groups corresponding to polarized $\mathbb{Q}$-Hodge structures. Basing on this consideration, we introduce the class of abelian Cousin groups and prove an analogue of Poincar\'e complete reducibility theorem for them.