${\mathbb{T}}^n$-invariant Kaehler-Einstein manifolds immersed in complex projective spaces

We give a complete list, for $n\le 6$, of non-isometric ${\mathbb{T}}^n$-invariant Kaehler-Einstein manifolds immersed in a finite dimensional complex projective space endowed with the Fubini-Study metric. This solves, in the aforementioned case, a classical and long-staying problem addressed among others by Calabi and Chern.