Bilinear forms and extremal Kähler vector fields associated with Kähler classes (Q1345920)

Let \(X\) be an \(n\)-dimensional compact complex connected manifold with a Kähler class \(\gamma \in H^{1,1} (X,\mathbb{R}) := H^{1,1} (X) \cap H^2(X,\mathbb{R})\). A bilinear form is constructed and the uniqueness of extremal Kähler vector fields is proved. By using \textit{A. Futaki} and \textit{T. Mabuchi} [Lect. Notes Math. 1339, 127-141 (1988; Zbl 0646.32015)] and \textit{T. Mabuchi} [Proc. Japan Acad., Ser. A 69, No. 10, 392-394 (1993; Zbl 0807.53027)] the periodicity of extremal Kähler vector fields for \(\gamma = c_1(X)\) is obtained.