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

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 π π . We also put all our technical results together to complete the proof of the main theorem.