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We consider actions of non-compact simple Lie groups preserving an analyticrigid geometric structure of algebraic type on a compact manifold. Thestructure is not assumed to be unimodular, so an invariant measure may notexist. Ergodic stationary measures always exist, and when such a measure hasfull support, we show the following.  1) Either the manifold admits a smooth equivariant map onto a homogeneousprojective variety, defined on an open dense conull invariant set, or the Liealgebra of the Zariski closure of the Gromov representation of the fundamentalgroup contains a Lie subalgebra isomorphic to the Lie algebra of the actinggroup. As a corollary, a smooth non-trivial homogeneous projective factor doesexist whenever the fundamental group of $M$ admits only virtually solvablelinear 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 ofcertain simple groups (of real rank one) may indeed fail to have a smoothprojective factor.  3) It is possible to generalize Gromov's theorem on the algebraic hull of therepresentation of the fundamental group of the manifold to the case of analyticrigid non-unimodular structures, for actions of simple groups of any real rank.  An important ingredient in the proofs is a generalization of Gromov'scentralizer theorem beyond the case of invariant measures.

Invariant rigid geometric structures and smooth projective factors