### 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 2π(1-α) for αε (0, 1). In this paper we study how the existence of such Kähler-Einstein metrics depends on α. 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 αε((n+1)/(n+2), 1), where n is the dimension. We also give a characterisation of the pairs that admit negatively curved cone-edge Kähler-Einstein metrics with cone angle close to 2π. Again if these metrics exist for all cone-angles close to 2π, 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.

Original language | English (US) |
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Pages (from-to) | 321-338 |

Number of pages | 18 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 159 |

Issue number | 2 |

DOIs | |

State | Published - Sep 4 2015 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Mathematical Proceedings of the Cambridge Philosophical Society*,

*159*(2), 321-338. https://doi.org/10.1017/S0305004115000377