Prescribing Ricci curvature on homogeneous spaces

The prescribed Ricci curvature problem in the context of G-invariant metrics on a homogeneous space M = G/K is studied. We focus on the metrics at which the map g bar right arrow Rc(g) is, locally, as injective and surjective as it can be. Our main result is that such property is generic in the compact case. Our main tool is a formula for the Lichnerowicz Laplacian we prove in terms of the moment map for the variety of algebras.