Kähler-Einstein metrics with edge singularities

Details Kähler-Einstein metrics with edge singularities Kähler-Einstein metrics with edge singularities

2011-05-26

arxiv.org/abs/1105.5216v2

Mathematics

Differential Geometry

52 pages. with an appendix by Chi Li and Yanir A. Rubinstein. Improved exposition and references

Scientific paper

This article considers the existence and regularity of \KE metrics on a compact \K manifold $M$ with edge singularities with cone angle $2\pi \be$ along a smooth divisor $D$. We prove existence of such metrics with negative, zero and some positive cases for all cone angles $2\pi \be \leq 2\pi$. The results in the positive case parallel those in the smooth case. We also establish that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coefficients along $D$ for all $2\pi \be < 2\pi$. This work rests on a recent advance by Donaldson \cite{D}; certain of the existence results overlap those in other recent articles \cite{Berm,Br,CGP}.

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