K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers

2025/12/03 by Charles Cifarelli, Cifarelli, Charles, Carlos Esparza +1
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2512.03323

Abstract

Recently, Sun-Zhang have developed an algebraic theory for Kähler-Ricci shrinkers showing that they admit the structure of a polarized Fano fibration (π: X → Y, ξ). In particular, they conjecture that existence of a Kähler-Ricci shrinker metric is equivalent to a notion of K-stability. We prove one direction of this conjecture, namely that existence of a Kähler-Ricci shrinker metric g implies K-polystability of (π: X → Y, ξ), in the case that the Ricci curvature of g decays at infinity.

Citations

• Shrinking gradient Kähler-Ricci solitons are simply-connected
• Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons
• Kähler-Ricci shrinkers and Fano fibrations
• Weighted K-stability for a class of non-compact toric fibrations
• On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations
• Algebraicity of the Metric Tangent Cones and Equivariant K-stability
• On equivariant Serre problem for principal bundles
• Conical structure for shrinking Ricci solitons
• K-Semistability for irregular Sasakian manifolds

Related

• Kähler-Ricci shrinkers and Fano fibrations
• K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
• Algebraicity of the Metric Tangent Cones and Equivariant K-stability
• K-stability and parabolic stability
• Asymptotic Chow polystability in Kähler geometry