Pseudo-Kähler metrics on six-dimensional nilpotent Lie algebras
The aim of the present paper is to classify the 6-dimensional nilpotent Lie algebras \({\mathfrak g}\) which admit pseudo-Kähler metrics and to construct explicit pseudo-Kähler metrics when they exist, extending previous results given in [\textit{L. A. Cordero}, \textit{M. Fernández} and \textit{L. Ugarte}, Commentat. Math. Univ. Carol. 43, No. 2, 215--229 (2002; Zbl 1078.53020)]. In the case of 4-dimensions, there is only one nilpotent Lie algebra admitting pseudo-Kähler structures, the one underlying the well-known Kodaira-Thurston manifold [\textit{W. P. Thurston}, Proc. Am. Math. Soc. 55, 467--468 (1976; Zbl 0324.53031)]. From the standpoint of \textit{S. M. Salamon} [J. Pure Appl. Algebra 157, No. 2--3, 311--333 (2001; Zbl 1020.17006)] on the existence of complex structures on 6-dimensional nilpotent Lie algebras \({\mathfrak g}\), the authors prove that the complex structure \(J\) underlying any pseudo-Kähler structure on \({\mathfrak g}\) must be nilpotent in the sense of \textit{L. A. Cordero}, \textit{M. Fernández}, \textit{A. Gray} and \textit{L. Ugarte} [Trans. Am. Math. Soc. 352, No. 12, 5405--5433 (2000; Zbl 0965.32026)]. Also, they complete some results of \textit{A. Fino}, \textit{M. Parton} and \textit{S. M. Salamon} [Comment. Math. Helv. 79, No. 2, 317--340 (2004; Zbl 1062.53062)], according to which, these metrics are Ricci-flat, although in many cases nonflat. Moreover, they study the variation of the curvature tensor along curves of pseudo-Kähler metrics.
scientific article; zbMATH DE number 2183087