Paper · Feb 19, 2025
Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons
As on arXiv
Abstract
We show that, up to biholomorphism, a given noncompact complex manifold only admits one shrinking gradient Kähler-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.
Comments: 50 pages; removed the hypothesis that the vertices of the cones coincide from Prop. 5.17 (new numbering). The numbers of results in section 5 have changed
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Shrinking gradient Kähler-Ricci solitons are simply-connected
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A Matsushima theorem, uniqueness and K-polystability for Kähler-Ricci shrinkers
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.