Curvature properties of twistor spaces of quaternionic Kähler manifolds (Q1266473)

The authors supply \((4n+2)\)-dimensional examples \((n>1)\) of strictly almost Kähler manifolds by studying natural almost Hermitian structures on twistor spaces of \(4n\)-dimensional quaternionic Kähler manifolds and show that some of them give negative answers to the questions raised by \textit{D. E. Blair} and \textit{S. Ianus} [Contemp. Math. 51, 23-29 (1986; Zbl 0591.53030)] and by \textit{A. Gray} [Tôhoku Math. J., II. Ser. 28, 601-612 (1976; Zbl 0351.53040)], respectively, of whether a compact almost Kähler manifold with Hermitian Ricci tensor or whose curvature tensor belongs to the class AH2 is Kähler. Negative answers to both of the questions were first given in dimension 6: to the first question by \textit{J. Davidov} and \textit{O. Muškarov} [Proc. Am. Math. Soc. 109, 1115-1120 (1990; Zbl 0704.53051)] and to the second question by \textit{J. Davidov}, \textit{O. Muškarov} and \textit{G. Grantcharov} [Ill. J. Math. 39, 627-634 (1995; Zbl 0844.53038)], respectively.