Characterizing rational homogeneous spaces via C^*-actions

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