Paper · Dec 3, 2025
K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers
As on arXiv
Abstract
Recently, Sun-Zhang have developed an algebraic theory for Kähler-Ricci shrinkers showing that they admit the structure of a polarized Fano fibration . 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 implies K-polystability of , in the case that the Ricci curvature of decays at infinity. As an application, we give a non-existence result: if is the blowup of a six-dimensional quadric along a two-dimensional subquadric, then the total space of the cube root of is a polarized Fano fibration not admitting a Kähler-Ricci shrinker.
Comments: Added section 7. 45 pages
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Background reading
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In the guidebook
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
- 11A.12Ricci Flow on Surfaces
Uniformization by Ricci flow, and Hamilton’s entropy.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.