Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 \pi

TL;DR

This paper studies the limits of Kahler-Einstein metrics with cone singularities on Fano manifolds when the cone angle is less than 2π, showing they correspond to projective algebraic varieties with standard cone singularities in smooth cases.

Contribution

It proves that Gromov-Hausdorff limits of such metrics are naturally projective algebraic varieties, extending understanding of metric limits with cone singularities.

Findings

Limits are projective algebraic varieties.

In smooth cases, limits have standard cone singularities.

Results extend previous work on Kahler-Einstein metrics with cone angles.

Abstract

This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. 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\pi. 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.