Numerical semistability of projective toric varieties
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917584756015104 |
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| author | Yotsutani, Naoto |
| author_facet | Yotsutani, Naoto |
| contents | Let $X \to \mathbb P^N$ be a smooth linearly normal projective variety. It was proved by Paul that the $K$-energy of $(X, {ω_{FS}}|_{X})$ restricted to the Bergman metrics is bounded from below if and only if the pair of (rescaled) Chow/Hurwitz forms of $X$ is numerically semistable. In this paper, we provide a necessary and sufficient condition for a given smooth toric variety $X_P$ to be numerically semistable with respect to $\mathcal O_{X_P}(i)$ for a positive integer $i$. Applying this result to a smooth polarized toric variety $(X_P, L_P)$, we prove that $(X_P, L_P)$ is asymptotically numerically semistable if and only if it is K-semistable for toric degenerations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02010 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical semistability of projective toric varieties Yotsutani, Naoto Algebraic Geometry Differential Geometry 51M20, 14M25, 53C55 Let $X \to \mathbb P^N$ be a smooth linearly normal projective variety. It was proved by Paul that the $K$-energy of $(X, {ω_{FS}}|_{X})$ restricted to the Bergman metrics is bounded from below if and only if the pair of (rescaled) Chow/Hurwitz forms of $X$ is numerically semistable. In this paper, we provide a necessary and sufficient condition for a given smooth toric variety $X_P$ to be numerically semistable with respect to $\mathcal O_{X_P}(i)$ for a positive integer $i$. Applying this result to a smooth polarized toric variety $(X_P, L_P)$, we prove that $(X_P, L_P)$ is asymptotically numerically semistable if and only if it is K-semistable for toric degenerations. |
| title | Numerical semistability of projective toric varieties |
| topic | Algebraic Geometry Differential Geometry 51M20, 14M25, 53C55 |
| url | https://arxiv.org/abs/2401.02010 |