Cheeger-Colding-Tian theory for conic Kähler-Einstein metrics (Q2659472)

In a series of papers written in the 1990s, Cheeger and Colding studied singular structures of spaces which arise as limits of sequences of Riemannian manifolds with Ricci curvature bounded below in the Gromov-Hausdorff topology. In particular, they proved the existence of tangent cones for the limiting space. Later, Cheeger, Colding and Tian gave further constraints on singularities of the limit space under a certain curvature condition for the approximating spaces. In the paper under review the authors extend the Cheeger-Colding-Tian theory to the so-called conic Kähler-Einstein metrics, a special case of conic Kähler metrics. In general, there are no smooth approximations of a family of conic Kähler-Einstein metrics with Ricci curvature uniformly bounded from below. So we have to deal with the technical issues to extend the original arguments. This extension provides a technical tool for a recent work of the authors of the paper under review and C. Li, in which they prove a version of the Yau-Tian-Donaldson conjecture for Fano varieties with a certain type of singularity [\textit{C. Li} et al., ``On Yau-Tian-Donaldson conjecture for singular Fano varieties'', Preprint, \url{arXiv:1711.09530}].