Einstein metrics on principal circle bundles (Q1385088)

Let \( P \) be a principal circle bundle with a connection over a compact Kähler manifold \(M\) equipped with a Riemannian metric \(g\) in the natural way. The author proves that if the metric \(g\) is Einstein and the curvature form of the connection has type \((1,1)\), then \(M\) is a product of Kähler-Einstein manifolds. A construction of an Einstein metric \(g\) in a circle bundle \(P\) over some product of Kähler-Einstein manifolds with known first Chern class was proposed by \textit{M. Y. Wang} and \textit{W. Ziller} [J. Differ. Geom. 31, 215-248 (1990; Zbl 0691.53036)] and the author's result establishes uniqueness of this construction.