Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than \boldmath2𝜋

Abstract

This is the second of a series of three papers which prove the fact that a <italic>K</italic> -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 is less than 2 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> π </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\pi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that these are in a natrual way projective algebraic varieties. In the case when the limiting variety and the limiting divisor are smooth we show that the limiting metric also has standard cone singularities.