The classification conjecture for projectively induced $S^1$-invariant Kähler–Einstein manifolds

An $S^1$-invariant Kähler manifold is a Kähler manifold admitting, in suitable local holomorphic coordinates, a Kähler potential depending only on $(|z_1|^2,\dots,|z_n|^2)$. A Kähler metric is projectively induced if it arises from a Kähler immersion into a complex projective space with the Fubini–Study metric. Let $g_{FS}$ denote the Fubini–Study metric, and let $n_1,\dots,n_k$ be positive integers. Classification conjecture. The only projectively induced and $S^1$-invariant Kähler–Einstein manifolds are open subsets of $\mathbb{C}\mathrm{P}^{n_1}\times\cdots\times\mathbb{C}\mathrm{P}^{n_k}$ endowed with the Kähler metric

$$q\left(c_1g_{FS}\oplus\cdots\oplus c_kg_{FS}\right),$$

where $k$ and $q\in\mathbb{Z}^+$, and, if $G$ is the least common multiple of $(n_1+1,\dots,n_k+1)$, then

$$c_i=\frac{G}{n_i+1}\quad\text{for }i=1,\dots,k.$$

This is the specialization of the preceding flag-manifold conjecture to the $S^1$-invariant setting, using that complex projective spaces are the only irreducible $S^1$-invariant flag manifolds and that only integer multiples of the Fubini–Study metric are projectively induced. The paper investigates this formulation, but the supplied text gives no resolution of the conjecture.