Toric Fano manifolds that do not admit extremal Kähler metrics

2024/11/26 by Hwang, DongSeon, Sato, Hiroshi, Yotsutani, Naoto
#14L24 #14M25 #53C55 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2411.17574

Abstract

We show that there exists a toric Fano manifold of dimension 10 that does not admit an extremal Kähler metric in the first Chern class, answering a question of Mabuchi. By taking a product with a suitable toric Fano manifold, one can also produce a toric Fano manifold of dimension n admitting no extremal Kähler metric in the first Chern class for each n ≥ 11.

Related

• Examples of non-symmetric Kähler-Einstein toric Fano manifolds
• Kahler geometry on toric manifolds, and some other manifolds with large symmetry
• Toric Fano manifolds of dimension at most eight with positive second Chern characters
• Generalized Kähler Einstein metrics and uniform stability for toric Fano manifolds
• Einstein-Kaehler Metrics on Symmetric Toric Fano Manifolds