Degenerating conic Kähler-Einstein metrics to the normal cone

Abstract

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

71 pages, v2: final version, to appear in Geometry & Topology