Characterizing rational homogeneous spaces via ${\mathbb{C}}^{\ast }$-actions

We study smooth varieties of Picard number one admitting a special dominating family of rational curves and an equalized ${\mathbb{C}}^{\ast }$-action. In particular we show that $X$ is a smooth variety of Picard number one with nef tangent bundle admitting an equalized ${\mathbb{C}}^{\ast }$-action with an isolated extremal fixed point if and only if $X$ is an irreducible Hermitian symmetric space.