In this paper, we give a criterion for the properness of the K-energy in a general Kähler class of a compact Kähler manifold by using Song-Weinkove's result in [24]. As applications, we give some Kähler classes on CP 2 #3CP 2 and CP 2 #8CP 2 in which the K-energy is proper. Finally, we prove Song-Weinkove's result on the existence of critical points ofĴ functional by the continuity method.

In this paper, we show that the α m,2 -invariant (introduced by Tian in [27] and [29]) of a smooth cubic surface with Eckardt points is strictly bigger than 2 3 . This can be used to simplify Tian's original proof of the existence of Kähler-Einstein metrics on such manifolds. We also sketch the computations on cubic surfaces with one ordinary double points, and outline the analytic difficulties to prove the existence of orbifold Kähler-Einstein metrics.Here the functions |s i | 2 , i = 0, . . . , N m are only defined locally by choosing a local trivialization of K −m X . But it's easy to see that the (1,1)-form on the right hand side is independent of the trivialization we choose and is globally defined.Definition 1.1 (Tian [24], [25]). The α-invariant and α m -invariant of X are defined to be:

We prove the existence of Kähler-Ricci solitons on toric Fano orbifolds, hence extend the theorem of Wang and Zhu (Adv Math 188:87-103, 2004) to the orbifold case.

In this short note, we prove the existence of constant scalar curvature Kähler metrics on compact Kähler manifolds with semi-ample canonical bundles.

We prove the expansion formula for the classical Futaki invariants on the blowup of Kähler surfaces, which explains the balancing condition of Arezzo-Pacard in [3]. The relation with Stoppa's result [18] is also discussed.