Kähler-Einstein toric submanifolds of the projective space
Antonio J. Di Scala, Martín Sombra
TL;DR
The work presents an algebraic framework to test whether Kähler-Einstein metrics on toric Fano manifolds can be realized as pullbacks of the Fubini-Study form under projective immersions. By translating geometry into Laurent-polynomial data via the δ and μ operators and Newton polytopes, the authors derive strict obstructions (GEC) and prove nonexistence results for the KE-immersion question on the symmetric families V_k, S_{m,k}, X_{m,k}, W_m and on two non-symmetric examples, thereby supporting a broader conjecture. A key corollary shows that centrally symmetric toric Fano manifolds admit a projectively induced KE metric if and only if they decompose as a product of projective lines. The methods provide a robust, coordinate-free algebraic approach to characterize projectively induced KE metrics in higher-dimensional toric settings, with potential implications for broader classifications.
Abstract
We show that the Kähler-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 Kähler-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 Kähler-Einstein metrics can be induced by projective immersions.