On the blow-up formula of weighted stability for polarized toric manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909698100297728 |
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| author | Lee, King Leung Yotsutani, Naoto |
| author_facet | Lee, King Leung Yotsutani, Naoto |
| contents | Let $X$ be a smooth projective toric variety, and let $\widetilde{X}$ denote the blow-up of $X$ at finitely many distinct tours-invariant points. This paper provides an explicit combinatorial formula for the Chow weight of $\widetilde{X}$ in terms of the base toric manifold $X$ and the symplectic cuts of its associated Delzant polytope. We apply this blow-up formula to the projective plane and compare Chow stability of toric blow-ups with that of blow-ups at general points. Furthermore, we derive the blow-up formula of the Futaki-Ono invariant, which serves an obstruction to asymptotic Chow semistability of a polarized toric manifold. In the final section, we extend our approach to study blow-up formulas for weighted (Chow/K-) stability using our combinatorial framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10082 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the blow-up formula of weighted stability for polarized toric manifolds Lee, King Leung Yotsutani, Naoto Algebraic Geometry Differential Geometry Symplectic Geometry 51M20, 53C55, 14M25 Let $X$ be a smooth projective toric variety, and let $\widetilde{X}$ denote the blow-up of $X$ at finitely many distinct tours-invariant points. This paper provides an explicit combinatorial formula for the Chow weight of $\widetilde{X}$ in terms of the base toric manifold $X$ and the symplectic cuts of its associated Delzant polytope. We apply this blow-up formula to the projective plane and compare Chow stability of toric blow-ups with that of blow-ups at general points. Furthermore, we derive the blow-up formula of the Futaki-Ono invariant, which serves an obstruction to asymptotic Chow semistability of a polarized toric manifold. In the final section, we extend our approach to study blow-up formulas for weighted (Chow/K-) stability using our combinatorial framework. |
| title | On the blow-up formula of weighted stability for polarized toric manifolds |
| topic | Algebraic Geometry Differential Geometry Symplectic Geometry 51M20, 53C55, 14M25 |
| url | https://arxiv.org/abs/2407.10082 |