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Uniqueness of asymptotically conical shrinking gradient K\"ahler-Ricci solitons
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We show that, up to biholomorphism, a given noncompact complex manifold only admits one shrinking gradient K\"ahler-Ricci soliton with Ricci curvature tending to zero at infinity. Our result does not require fixing the asymptotic data of the metric, nor fixing the soliton vector field. The method used to prove the uniqueness of the soliton vector field can be applied more widely, for example to show that conical Calabi-Yau metrics on a given complex manifold are unique up to biholomorphism. We also use it to prove that if two polarized Fano fibrations, as introduced by Sun-Zhang, are biholomorphic, then they are isomorphic as algebraic varieties.
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Cited by 3 Pith papers
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Examples of complete Calabi--Yau metrics on affine smoothings of irregular toric Calabi--Yau cones
New infinite families of affine Calabi-Yau manifolds with irregular toric tangent cones are constructed, with an explicit algorithm for the Reeb field and Minkowski decompositions.
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Uniqueness of shrinking K\"ahler-Ricci solitons on resolutions of K\"ahler cones
Every complete shrinking gradient Kähler-Ricci soliton on a resolution of a Kähler cone is asymptotically conical, which yields uniqueness up to pullback by biholomorphism.
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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.
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