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Remarks on Bott residue formula and Futaki-Morita integral invariants

机译：关于Bott残差公式和Futaki-Morita积分不变量的说明

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摘要

When a compact complex manifold admits a non-degenerate holomorphic vector field, the famous Bott residue formula reduces the calculations of Chern numbers to the zero set of this vector field. The Futaki invariant obstructs the existence of Kahler-Einstein metric with positive scalar curvature. Inspired by the proof of Bott residue formula, Futaki and Morita defined a family of integral invariants, which include Futaki's original invariant as a special case, and gave them corresponding residue formulae which have the same feature as that of Bott. They also proved some properties of these integral invariants when the underlying manifolds are Kahler. We remark that some considerations of Futaki and Morita on these integral invariants are closely related to some much earlier literatures and recent work of the author. The purpose of this paper is to generalize some considerations of them and give some new properties of these integral invariants. Some related remarks and articles are also discussed in this note.
机译：当一个紧凑的复流形接受一个非退化的全纯矢量场时，著名的波特残差公式将切恩数的计算减少到该矢量场的零集。 Futaki不变量阻碍标量曲率为正的Kahler-Einstein度量的存在。受Bott残差公式证明的启发，Futaki和Morita定义了一个积分不变式族，其中包括Futaki的原始不变量作为特例，并为其提供了与Bott具有相同特征的相应残差公式。他们还证明了当基础流形为Kahler时这些积分不变量的某些性质。我们注意到，Futaki和Morita对这些积分不变式的某些考虑与一些较早的文献和作者的近期工作紧密相关。本文的目的是概括它们的一些考虑，并为这些积分不变量提供一些新的性质。本说明中还讨论了一些相关的评论和文章。

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