3 problems

Folklore conjecture. Then $(X,g)$ is a homogeneous space.

Let $X$ be a Fano manifold with volume form $dV$, let $\beta_0<0$, and let $\beta$ range over the real numbers satisfying $\beta>\beta_0$. For each $N$, let $\mathcal{Z}_N(\beta)$…

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.