A new first-order visibility argument derives compactness, splitting, and Gaussian rigidity for complete Kähler-Ricci shrinkers from the weight structure of their polarized Fano fibrations.
Uniqueness of shrinking K\"ahler-Ricci solitons on resolutions of K\"ahler cones
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abstract
We show that any complete shrinking gradient K\"ahler-Ricci soliton on a resolution of a K\"ahler cone is necessarily asymptotically conical. From a result of Esparza, it then follows that up to pullback by biholomorphism, there exists at most one complete shrinking gradient K\"ahler-Ricci soliton on such a resolution. This confirms a special case of the uniqueness part of a conjecture of Song-Zhang. Some other consequences are also discussed.
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math.DG 1years
2026 1verdicts
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Compactness and Rigidity of Complete K\"ahler Ricci Shrinkers
A new first-order visibility argument derives compactness, splitting, and Gaussian rigidity for complete Kähler-Ricci shrinkers from the weight structure of their polarized Fano fibrations.