Song Sun

Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities

This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.

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

This is the third and final paper in a series which establish 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 approaches 2\pi. We also put all our technical results together to complete the proof of the main theorem that if a K-stable Fano manifold admits a Kahler-Einstein metric.

Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2

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.

Space of Kähler Metrics (V) – Kähler Quantization

Given a polarized Kahler manifold (X,L). The space ℋ of Kahler metrics in\( 2_\Pi {c_1}{(L)}\)is an infinite-dimensional Riemannian symmetric space. As a metric space, it has non-positive curvature. There is associated to ℋ a sequence of finite-dimensional symmetric spaces\( \mathcal{B}_{k}{({k}\, \epsilon \,\mathbb{N})} \) of non-compact Type. We prove that ℋ is the limit of \( \mathcal{B}_{k} \)as metric spaces in certain sense. As applications, this provides more geometric proofs of certain known geometric properties of the space ℋ.