( de Gruyter 2003 The Ricci tensor of an almost homogeneous Kähler manifold

Abstract. We determine an explicit expression for the Ricci tensor of a K-manifold, that is of a compact Kähler manifoldM with vanishing first Betti number, on which a semisimple group G of biholomorphic isometries acts with an orbit of codimension one. We also prove that, up to few exceptions, the Kähler form o and the Ricci form r of a K-manifold M are uniquely determined by two special curves with values in g LieðGÞ, say Zo;Zr: R! g, and we show how Zr is determined by Zo. These results are used in another work with F. Podestà, where new examples of non-homogeneous compactKähler–Einsteinmanifoldswith positive first Chern class are constructed. Key words. Kähler–Einstein metrics, cohomogeneity one actions. 2002 Mathematics Subject Classification. 53C55, 53C25, 57S15