K-stable valuations and Calabi-Yau metrics on affine spherical varieties
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| Format: | Preprint |
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2024
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| _version_ | 1866910829536870400 |
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| author | Nghiem, Tran-Trung |
| author_facet | Nghiem, Tran-Trung |
| contents | 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, $K$ 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 other 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05833 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | K-stable valuations and Calabi-Yau metrics on affine spherical varieties Nghiem, Tran-Trung Algebraic Geometry Differential Geometry 53C25, 53C55, 32Q25, 14M27 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, $K$ 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 other 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. |
| title | K-stable valuations and Calabi-Yau metrics on affine spherical varieties |
| topic | Algebraic Geometry Differential Geometry 53C25, 53C55, 32Q25, 14M27 |
| url | https://arxiv.org/abs/2405.05833 |