Through the work of Cheeger, Colding, Naber and others we have a deep understanding of the structure of Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature lower bounds. For polarized Kahler manifolds, this was taken further by Donaldson-Sun, who showed that under two-sided Ricci curvature bounds, non-collapsed limit spaces are projective varieties, leading to major progress in Kahler geometry. I will discuss joint work with Gang Liu giving an extension of this result to the case when the Ricci curvature is only bounded from below.