Remarks on Bott residue formula and Futaki–Morita integral invariants

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 Kähler–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 Futakiʼs 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 Kähler. 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.