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| Format: | Preprint |
| Publié: |
2024
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2409.15718 |
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| _version_ | 1866912643240951808 |
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| author | Wang, Linsheng |
| author_facet | Wang, Linsheng |
| contents | We prove a generalization of the algebraic version of Tian conjecture. Precisely, for any smooth strictly increasing function $g:\mathbb{R}\to\mathbb{R}_{>0}$ with ${\rm log}\circ g$ convex, we define the $\mathbf{H}^g$-invariant on a Fano variety $X$ generalizing the $\mathbf{H}$-invariant introduced by Tian-Zhang-Zhang-Zhu, and show that $\mathbf{H}^g$ admits a unique minimizer. Such a minimizer will induce the $g$-optimal degeneration of the Fano variety $X$, whose limit space admits a $g'$-soliton. We present an example of Fano threefold which has the same $g$-optimal degenerations for any $g$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15718 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized optimal degenerations of Fano varieties Wang, Linsheng Algebraic Geometry We prove a generalization of the algebraic version of Tian conjecture. Precisely, for any smooth strictly increasing function $g:\mathbb{R}\to\mathbb{R}_{>0}$ with ${\rm log}\circ g$ convex, we define the $\mathbf{H}^g$-invariant on a Fano variety $X$ generalizing the $\mathbf{H}$-invariant introduced by Tian-Zhang-Zhang-Zhu, and show that $\mathbf{H}^g$ admits a unique minimizer. Such a minimizer will induce the $g$-optimal degeneration of the Fano variety $X$, whose limit space admits a $g'$-soliton. We present an example of Fano threefold which has the same $g$-optimal degenerations for any $g$. |
| title | Generalized optimal degenerations of Fano varieties |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2409.15718 |