Toric geometry of generalized Kähler-Ricci solitons

2025/09/01 by Vestislav Apostolov, Giuseppe Bárbaro, Apostolov, Vestislav +5
Mathematics · #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2509.01639

openalex publication_date 2025/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish a local equivalence between toric steady Kähler-Ricci solitons and A-type toric generalized Kähler-Ricci solitons (GKRS). Under natural global conditions we show this equivalence extends to complete GKRS, yielding a general construction of new examples in all dimensions. We show that in four dimensions, all GKRS are either described by the generalized Kähler Gibbons-Hawking ansatz, or have split tangent bundle, or are A-type toric. This yields a local classification in four dimensions, together with a conjecturally exhaustive construction of complete symplectic-type examples.

Citations

• Explicit complete Ricci-flat metrics and Kähler-Ricci solitons on direct sum bundles
• Rigidity results for non-Kähler Calabi-Yau geometries on threefolds
• Formal structure of scalar curvature in generalized Kähler geometry
• A family of Kähler flying wing steady Ricci solitons
• Hamiltonian 2-forms and new explicit Calabi--Yau metrics and gradient steady Kähler--Ricci solitons on ℂⁿ
• The generalized Kähler Calabi-Yau problem
• The Kähler geometry of toric manifolds
• On geometry of steady toric Kähler-Ricci solitons
• Variational structure and uniqueness of generalized Kähler-Ricci solitons
• Generalized scalar curvature and modified moment map in generalized Kähler geometry
• Generalized Kähler-Ricci flow on toric Fano varieties
• Asymptotically cylindrical steady Kähler-Ricci solitons
• Uniqueness of shrinking gradient Kähler-Ricci solitons on non-compact toric manifolds
• Generalized Ricci Flow
• Steady gradient Kähler-Ricci solitons on crepant resolutions of Calabi-Yau cones
• Extremal Kähler Poincaré type metrics on toric varieties
• The nondegenerate generalized Kähler Calabi-Yau problem
• Hessian metrics, CD(K,N)-spaces, and optimal transportation of\n log-concave measures
• Superconformal Structures on Generalized Calabi-Yau Metric Manifolds
• Constructing Kähler-Ricci solitons from Sasaki-Einstein manifolds
• A parabolic flow of pluriclosed metrics
• Kahler geometry on toric manifolds, and some other manifolds with large symmetry
• On Ricci solitons of cohomogeneity one
• Uniqueness of the Ricci Flow on Complete Noncompact Manifolds
• Interior estimates for solutions of Abreu's equation
• Hamiltonian 2-forms in Kahler geometry, II Global Classification
• Generalized complex geometry
• Kahler metrics on toric orbifolds

Related

• Toric Generalized Kähler-Ricci Solitons with Hamiltonian 2-form
• Classification of generalized Kähler-Ricci solitons on complex surfaces
• Special Kähler-Ricci potentials and Ricci solitons
• Kähler Soliton Surfaces Are Generically Toric
• Variational structure and uniqueness of generalized Kähler-Ricci solitons