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Pré-Publication, Document De Travail Année : 2024

K-stable valuations and Calabi-Yau metrics on affine spherical varieties

Valuations K-stables et métriques de Calabi-Yau sur les variétés affines sphériques

Résumé

After providing an explicit K-stability condition for a $\mathbb{Q}$-Gorenstein log spherical cone, we prove the existence and uniqueness of an equivariant K-stable degeneration of the cone, and deduce uniqueness of the asymptotic cone of a given complete $K$-invariant Calabi-Yau metric in the trivial class of an affine $G$-spherical manifold with $K$ being the maximal compact subgroup of $G$. Next, we prove that the valuation induced by $K$-invariant Calabi-Yau metrics on affine $G$-spherical manifolds is in fact $G$-invariant. As an application, we point out an affine smoothing of a Calabi-Yau cone that does not admit any $K$-invariant Calabi-Yau metrics asymptotic to the cone. Another corollary is that on $\mathbb{C}^3$, there are no complete Calabi-Yau metrics with maximal volume growth and spherical symmetry other than the standard flat metric and the Li-Conlon-Rochon-Székelyhidi metrics with horospherical asymptotic cone. This answers the question whether there is a nontrivial asymptotic cone with smooth cross section on $\mathbb{C}^3$ raised by Conlon-Rochon when the symmetry is spherical.
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Dates et versions

hal-04602819 , version 1 (06-06-2024)

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Identifiants

  • HAL Id : hal-04602819 , version 1

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Tran-Trung Nghiem. K-stable valuations and Calabi-Yau metrics on affine spherical varieties. 2024. ⟨hal-04602819⟩
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