Einstein metric on complex manifolds and the lifted Futaki invariant

点击翻译按钮获取中文摘要

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].