Einstein metric on complex manifolds and the lifted Futaki invariant

设M是一个封闭复流形。在M.In . K.Tsuboi，提升的Futaki不变量和自旋-狄拉克算子，大阪j .数学。， vol. 32(1995), 207-225，我们得到了一个计算提升的Futaki不变量的公式，它是Futaki不变量的推广。在下一页的[1]中，我们推广了这个公式，得到了几乎复杂流形的不动点公式。在[2]中，我们证明了某一线束的完整性是爱因斯坦-卡勒度规存在的障碍。常数标量Kahler度规是Einstein-Kahler度规的推广。在[3]，[4]中，研究了阻碍常数标量曲率Kahler度量存在的积分不变量Bando-Calabi-Futaki不变量与其他几何不变量的关系。为了得到关于积分不变量的结果，我们需要知道积分方程，这在[5]，[6]中有研究。

Let M be a closed complex manifold. Then the Futaki invariant is an obstruction to the existence of the Einstein-Kahler metric on M.In K.Tsuboi, The lifted Futaki invariants and the spinc-Dirac operators, Osaka J.Math., vol. 32 (1995), 207-225, we obtain a formula to calculate the lifted Futaki invariant, which is a generalization of the Futaki invariant. In [1] (of the next page), we generalize this formula and obtain a fixed point formula for almost complex manifolds. In [2], we show that the holonomy of a certain line bundle is an obstruction to the existence of the Einstein-Kahler metric. The constant scalar Kahler metric is a generalization of the Einstein-Kahler metric. In [3], [4], the relation of the Bando-Calabi-Futaki invariant, which is an obstruction to the existence of the constant scalar curvature Kahler metric and is an integral invariant, to other geometric invariants are studied. In order to obtain the result about the integral invariant, we need to know about the integral equation, which are studied in [5], [6].