Invariant pseudo-Kähler metrics on generalized flag manifolds
A pseudo-Riemannian manifold \((M,g)\) with a complex structure \(J\) is called a \textit{pseudo-Kähler manifold} if \(g\) is a pseudo-Hermitian metric and the fundamental \(2\)-form \(\omega _{g}\) is closed. Hence, a pseudo-Kähler manifold is one of the natural generalizations of a Kähler manifold. Such a pseudo-Riemannian metric \(g\) is called \textit{pseudo-Kähler metric} and the fundamental \(2\)-form \(\omega _{g}\) \textit{pseudo-Kähler structure}. In this interesting paper, the author considers signatures of invariant pseudo-Kähler metrics on generalized flag manifolds from the viewpoint of \(T\)-root systems. By a result of Maschler, to investigate the signatures of the integral pseudo-Kähler metrics of a generalized flag manifold, it is sufficient to investigate the signatures of the invariant pseudo-Kähler metrics. By investigating patterns of signatures of invariant pseudo-Kähler metrics on a generalized flag manifold, the author gets that two invariant complex structures \(J\) and \(J^{\prime }\) on a generalized flag manifold are either diffeomorphic or non-diffeomorphic.
scientific article; zbMATH DE number 6351463