We show that if a compact Kaehler manifold $M$ admits closed Fedosov's star product then the reduced Lie algebra of holomorphic vector fields on $M$ is reductive. This comes in pair with the obstruction previously found by La Fuente-Gravy. More generally we consider the squared norm of Cahen-Gutt moment map as in the same spirit of Calabi functional for the scalar curvature in cscK problem, and prove a Cahen-Gutt version of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kaehler manifolds. The proof uses a Hessian formula for the squared norm of Cahen-Gutt moment map.

中文翻译：

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Cahen-Gutt矩图、闭Fedosov星积和自同构群结构

我们证明，如果紧致的 Kaehler 流形 $M$ 承认闭 Fedosov 的星积，那么 $M$ 上全纯向量场的约化李代数是约简的。这与 La Fuente-Gravy 先前发现的障碍物相结合。更一般地，我们考虑 Cahen-Gutt 矩图的平方范数，与 CscK 问题中标量曲率的 Calabi 泛函精神相同，并证明关于全纯向量场的李代数结构的 Calabi 定理的 Cahen-Gutt 版本对于极值 Kaehler 流形。证明使用 Hessian 公式计算 Cahen-Gutt 矩图的平方范数。