Positivity in Kähler-Einstein theory

Abstract

Tian initiated the study of incomplete Kähler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle $2\pi (1-\alpha )$ for $\alpha \in (0,1)$. In this paper we study how the existence of such Kähler-Einstein metrics depends on $\alpha$. We show that in the negative scalar curvature case, if such Kähler-Einstein metrics exist for all small cone-angles then they exist for every $\alpha \in (\frac{n+1}{n+2},1)$, where $n$ is the dimension. We also give a characterization of the pairs that admit negatively curved cone-edge Kähler-Einstein metrics with cone angle close to $2\pi$. Again if these metrics exist for all cone-angles close to $2\pi$, then they exist in a uniform interval of angles depending on the dimension only. Finally, we show how in the positive scalar curvature case the existence of such uniform bounds is obstructed.

arXiv commentSome changes according the comments of the referee and references updated