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Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches $ 2\pi$ and completion of the main proof


Authors: Xiuxiong Chen, Simon Donaldson and Song Sun
Journal: J. Amer. Math. Soc. 28 (2015), 235-278
MSC (2010): Primary 53C55
Published electronically: March 28, 2014
MathSciNet review: 3264768
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Abstract: This is the third and final article in a series which prove the fact that a K-stable Fano manifold admits a Kähler-Einstein metric. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2$ \pi $. We also put all our technical results together to complete the proof of the main theorem.


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Xiuxiong Chen
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651 – and – School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China
Email: xiu@math.sunysb.edu

Simon Donaldson
Affiliation: Department of Mathematics, Imperial College London, London, U.K.
Email: s.donaldson@imperial.ac.uk

Song Sun
Affiliation: Department of Mathematics, Imperial College London, London, U.K.
Email: s.sun@imperial.ac.uk

DOI: https://doi.org/10.1090/S0894-0347-2014-00801-8
Received by editor(s): March 8, 2013
Received by editor(s) in revised form: October 4, 2013, and January 13, 2014
Published electronically: March 28, 2014
Additional Notes: The first author was partly supported by National Science Foundation grant No 1211652; the last two authors were partly supported by the European Research Council award No 247331.
Article copyright: © Copyright 2014 American Mathematical Society