The flag-manifold conjecture for projectively immersed Kähler–Einstein manifolds

A Kähler–Einstein manifold is a Kähler manifold whose Ricci tensor is proportional to its metric, and a Kähler immersion is a holomorphic isometric immersion into a complex space form. A complex flag manifold here means a compact simply-connected Kähler manifold acted upon transitively by its holomorphic isometry group. Flag-manifold conjecture. A Kähler–Einstein manifold that can be Kähler immersed into a complex projective space is an open subset of a complex flag manifold. The conjecture concerns the classification of Kähler–Einstein manifolds admitting Kähler immersions into complex projective space; the paper tests it for $S^1$-invariant Kähler manifolds, while the general classification remains open.