Anticanonically balanced metrics and the Hilbert-Mumford criterion for the $δ_m$-invariant of Fujita-Odaka
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911759027142656 |
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| author | Hashimoto, Yoshinori |
| author_facet | Hashimoto, Yoshinori |
| contents | We prove that the stability condition for Fano manifolds defined by Saito-Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the result by Rubinstein-Tian-Zhang, we obtain the following algebro-geometric corollary: the $δ_m$-invariant of Fujita-Odaka satisfies $δ_m >1$ if and only if the Fano manifold is stable in the sense of Saito-Takahashi, establishing a Hilbert-Mumford type criterion for $δ_m >1$. We also extend this result to the Kähler-Ricci $g$-solitons and the coupled Kähler-Einstein metrics, and as a by-product we obtain a formula for the asymptotic slope of the coupled Ding functional in terms of multiple test configurations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_12346 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Anticanonically balanced metrics and the Hilbert-Mumford criterion for the $δ_m$-invariant of Fujita-Odaka Hashimoto, Yoshinori Differential Geometry Algebraic Geometry 53C55 (Primary), 32Q26 (Secondary) We prove that the stability condition for Fano manifolds defined by Saito-Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the result by Rubinstein-Tian-Zhang, we obtain the following algebro-geometric corollary: the $δ_m$-invariant of Fujita-Odaka satisfies $δ_m >1$ if and only if the Fano manifold is stable in the sense of Saito-Takahashi, establishing a Hilbert-Mumford type criterion for $δ_m >1$. We also extend this result to the Kähler-Ricci $g$-solitons and the coupled Kähler-Einstein metrics, and as a by-product we obtain a formula for the asymptotic slope of the coupled Ding functional in terms of multiple test configurations. |
| title | Anticanonically balanced metrics and the Hilbert-Mumford criterion for the $δ_m$-invariant of Fujita-Odaka |
| topic | Differential Geometry Algebraic Geometry 53C55 (Primary), 32Q26 (Secondary) |
| url | https://arxiv.org/abs/2104.12346 |