Invariant rigid geometric structures and smooth projective factors

TL;DR

This paper investigates actions of non-compact simple Lie groups on compact manifolds with rigid geometric structures, revealing conditions for the existence of smooth projective factors and generalizing Gromov's theorem to non-unimodular cases.

Contribution

It introduces new criteria for the existence of smooth projective factors in rigid geometric structures and extends Gromov's algebraic hull theorem to non-unimodular settings.

Findings

Existence of smooth equivariant maps onto homogeneous projective varieties under certain conditions.

Construction of examples where no smooth projective factor exists for some simple group actions.

Generalization of Gromov's theorem to analytic rigid non-unimodular structures for all real ranks.

Abstract

We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show the following. 1) Either the manifold admits a smooth equivariant map onto a homogeneous projective variety, defined on an open dense conull invariant set, or the Lie algebra of the Zariski closure of the Gromov representation of the fundamental group contains a Lie subalgebra isomorphic to the Lie algebra of the acting group. As a corollary, a smooth non-trivial homogeneous projective factor does exist whenever the fundamental group of $M$ admits only virtually solvable linear representations, and thus in particular when $M$ is simply connected, regardless of the real rank. 2) There exist explicit examples showing that analytic rigid actions of certain simple groups (of real rank one) may indeed fail to have a smooth projective factor. 3) It is possible to generalize Gromov's theorem on the algebraic hull of the representation of the fundamental group of the manifold to the case of analytic rigid non-unimodular structures, for actions of simple groups of any real rank. An important ingredient in the proofs is a generalization of Gromov's centralizer theorem beyond the case of invariant measures.