The projectively induced Kähler–Einstein flag-manifold conjecture

A connected complex manifold $M$ is equipped with a Kähler–Einstein metric $g$. The metric is projectively induced when $M$ admits a Kähler immersion into a finite-dimensional complex projective space with its Fubini–Study metric. A complex flag manifold means a compact simply-connected Kähler manifold acted upon transitively by its holomorphic isometry group.

Flag-manifold conjecture. If $(M,g)$ is a Kähler–Einstein manifold endowed with a projectively induced metric, then it is an open subset of a complex flag manifold.

The conjecture proposes that finite-dimensional projectively induced Kähler–Einstein manifolds must arise locally from homogeneous flag manifolds. The paper tests this conjecture for rotation-invariant metrics; its validity in the stated generality remains open, while the infinite-dimensional analogue is explicitly known to fail.