K\"ahler-Einstein toric submanifolds of the projective space

TL;DR

This paper investigates which K"ahler-Einstein metrics on toric Fano manifolds can be realized through projective immersions, showing that only products of projective lines admit such metrics, supporting a broader conjecture.

Contribution

It demonstrates that most symmetric and non-symmetric toric Fano manifolds' K"ahler-Einstein metrics cannot be induced by projective immersions, except for products of projective lines.

Findings

Most symmetric toric Fano manifolds' K"ahler-Einstein metrics are not induced by projective immersions.

Non-symmetric examples also do not admit such induced metrics.

Only products of projective lines among centrally symmetric toric Fano manifolds admit induced K"ahler-Einstein metrics.

Abstract

We show that the K\"ahler-Einstein metrics on the four families of examples of symmetric toric Fano manifolds presented by Batyrev and Selivanova cannot be realized as metrics induced by immersions into projective spaces equipped with Fubini-Study metrics. We obtain a similar conclusion for the non-symmetric examples discovered by Nill and Paffenholz. A consequence is that a centrally symmetric toric Fano manifold admits a K\"ahler-Einstein metric induced by a projective immersion if and only if it is a product of projective lines. These results provide evidence for a broader conjecture characterizing which K\"ahler-Einstein metrics can be induced by projective immersions.