The main purpose of this paper is to prove a general result about the geometry of holomorphic line bundles over Khler manifolds. This result is essentially a partial verification of a conjecture of Tian [30] and Tian has, over many years, highlighted the importance of the question for the existence theory of KhlerEinstein metrics. We will begin by stating this main result.

This is the first of a series of three papers which prove the fact that a K-stable Fano manifold admits a Khler-Einstein metric. The main result of this paper is that a Khler-Einstein metric with cone singularities along a divisor can be approximated by a sequence of smooth Khler metrics with controlled geometry in the Gromov-Hausdorff sense.

This is the third and final article in a series which prove the fact that a K-stable Fano manifold admits a Khler-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.

We prove that the first Chern form of the moduli space of polarized Calabi-Yau manifolds, with the Hodge metric or the Weil-Petersson metric, represent the first Chern class of the canonical extensions of the tangent bundle to the compactification of the moduli space with normal crossing divisors.

In this paper, we solve a problem of Kobayashi posed in (Complex Finsler vector bundles, American Mathematical Society, Providence, 1996) by introducing a Donaldson type functional on the space F + ( E ) of strongly pseudo-convex complex Finsler metrics on <i>E</i>a holomorphic vector bundle over a closed Khler manifold <i>M</i>. This Donaldson type functional is a generalization in the complex Finsler geometry setting of the original Donaldson functional and has FinslerEinstein metrics on <i>E</i> as its only critical points, at which this functional attains the absolute minimum.