Kähler–Einstein Toric Submanifolds of the Projective Space

We show that the Kähler–Einstein metrics on the four families of symmetric toric Fano manifolds due to Batyrev and Selivanova cannot be induced by immersions into projective spaces equipped with the Fubini–Study metric. We also show a similar conclusion for the non-symmetric examples discovered by Nill and Paffenholz. As an application we obtain that a centrally symmetric toric Fano manifold admits a Kähler–Einstein metric induced by a projective immersion if and only if it is a product of projective lines. These results confirm for each of these examples a broader conjecture characterizing the Kähler–Einstein metrics that can be induced by projective immersions.