On Sun-Zhang's theory of Fano fibrations -- weighted volumes, moduli and bubbling Fano fibrations

2025/06/17 by Yuji Odaka, Odaka, Yuji
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2506.14671

openalex publication_date 2025/06/17 · openalex created_date 2025/10/18 · openalex updated_date 2026/08/01

Abstract

We revisit the recent theory of Sun-Zhang on general Fano fibration (germs) which emerged from the study of non-compact Kahler-Ricci soliton metrics, primarily from an algebro-geometric perspective. In addition to reviewing the existing framework, we present new results, conjectures, and remarks. These include methods for computing weighted volumes via (restricted) volumes, Laplace transforms, and incomplete Gamma-functions, and a conjectural algebro-geometric construction (``bubbling") of Fano fibration with asymptotically conical base from degenerating Fano fibration.

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Related

• Kähler-Ricci shrinkers and Fano fibrations
• Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow
• Fano Fibrations and Twisted Kähler-Einstein Metrics I
• The moduli space of Fano manifolds with Kähler-Ricci solitons
• Deformations of Fano manifolds with weighted solitons