Let \(M\) be a compact connected \(n\)-dimensional complex manifold, \(L\) a holomorphic line bundle over \(M\), \(p_i : M \times M \to M\) \(i = 1,2\) the natural projections. Put \(\widehat L = p_1^* L \otimes p^*_2 L^{-1}\) and consider the holomorphic vector bundle \(E(M, L) : = 1_{M \times M} \oplus 1_{M \times M} \otimes \widehat L\), \(1_{M \times M} : = M \times M \times \mathbb{C}\). Let \(\mathbb{P} (E(M,L))\) be the \(\mathbb{P}^2 (\mathbb{C})\)-bundle over \(M \times M\) associated to \(E(M,L)\). \(\mathbb{P} (E(M,L))\) has three natural cross-sections \(\Sigma_1, \Sigma_2, \Sigma_3\) corresponding to the three direct summands \(\{0\} \oplus \{0\} \oplus \widehat L\), \(\{0\} \oplus 1_{M \times M} \oplus \{0\}\), \(1_{M \times M} \oplus \{0\} \oplus \{0\}\). Blowing up \(\Sigma_1, \Sigma_2, \Sigma_3\) we get a complex manifold \({\mathfrak X} (M,L)\). — From now on \(M\) is a simply connected homogeneous space with a Kähler metric. A classical result states that \(M \simeq G/U\), where \(G\) is a simply connected complex semisimple Lie group and \(U\) is one of its parabolic subgroups. It follows easily that if \(L\) is a holomorphic line bundle over \(G/V\), \(G\) acts naturally on \(L\) inducing a \((G \times G)\)-action on \({\mathfrak X} (G/U,L)\). The author proves that the complex manifold \({\mathfrak X} (G/U,L)\) admits an Einstein-Kähler metric provided its first Chern class is positive.
Let now \(H\) be the hyperplane line bundle over \(\mathbb{P}^n (\mathbb{C})\). From the previous result follows that the toric Fano \((2n + 2)\)-fold \({\mathfrak X} (\mathbb{P}^n (\mathbb{C}), H^m)\), \(m \leq n\) admits an Einstein-Kähler metric, and in particular this allows to complete the classification of the toric Fano fourfolds given by the author in a previous paper.