Degenerating conic Kähler-Einstein metrics to the normal cone

Let $X$ be a Fano manifold of dimension at least $2$ and $D$ be a smooth divisor in a multiple of the anticanonical class, $\frac{1}{\alpha }(-K_X)$ with $\alpha >1$. It is well-known that Kähler-Einstein metrics on $X$ with conic singularities along $D$ may exist only if the angle $2\pi \beta$ is bigger than some positive limit value $2\pi {\beta }_{\ast }$. Under the hypothesis that the automorphisms of $D$ are induced by the automorphisms of the pair $(X,D)$, we prove that for $\beta >{\beta }_{\ast }$ close enough to ${\beta }_{\ast }$, such Kähler-Einstein metrics do exist. We identify the limits at various scales when $\beta \to {\beta }_{\ast }$ and, in particular, we exhibit the appearance of the Tian-Yau metric of $X\setminus D$.