Kähler-Einstein toric submanifolds of the projective space

2025/12/03 by Di Scala, Antonio J., Sombra, Martín
#32Q20 #53C24 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Primary 53C55 #Secondary 14M25

paper · doi:10.48550/arxiv.2512.03617

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.

Citations

• Kähler immersions of Kähler manifolds into complex space forms
• Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
• On Homothetic Balanced Metrics
• Kahler geometry on toric manifolds, and some other manifolds with large symmetry
• An algorithm for the classification of smooth Fano polytopes
• Einstein-Kaehler Metrics on Symmetric Toric Fano Manifolds

Related

• Projectively induced rotation invariant Kähler metrics
• Examples of non-symmetric Kähler-Einstein toric Fano manifolds
• Kähler-Einstein metrics
• Kahler geometry on toric manifolds, and some other manifolds with large symmetry
• Kähler-Einstein metrics along the smooth continuity method