Soliton-type metrics associated with weighted CSCK metrics on Fano manifolds
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912882951716864 |
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| author | Nakamura, Satoshi |
| author_facet | Nakamura, Satoshi |
| contents | We study weighted constant scalar curvature Kähler metrics, introduced by Lahdili as $(v,w)$-CSCK metrics, on Fano manifolds and their relationship with soliton-type metrics. In this paper, we introduce a weight function $g(v,w)$ associated with a pair of weight functions $(v,w)$. Assuming that $v$ and $g(v,w)$ are positive and log-concave on the moment polytope, we prove that the existence of a $(v,w)$-CSCK metric in the first Chern class is equivalent to the existence of a $g(v,w)$-soliton.
We also explain that a $g(v,w)$-soliton arises naturally from Sasaki geometry. More precisely, let $(v,w)$ be the weight functions defining a weighted CSCK metric in $2πc_1(X)$ which gives rise to a $\hatξ$-transverse extremal metric on an $S^1$-bundle $N$ in the canonical bundle of a Fano manifold $X$, where $\hatξ$ is a possibly irregular Reeb field on $N$. We prove that the associated $g(v,w)$-soliton on $X$ gives rise to a $\hatξ$-transverse Mabuchi soliton on $N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06306 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Soliton-type metrics associated with weighted CSCK metrics on Fano manifolds Nakamura, Satoshi Differential Geometry We study weighted constant scalar curvature Kähler metrics, introduced by Lahdili as $(v,w)$-CSCK metrics, on Fano manifolds and their relationship with soliton-type metrics. In this paper, we introduce a weight function $g(v,w)$ associated with a pair of weight functions $(v,w)$. Assuming that $v$ and $g(v,w)$ are positive and log-concave on the moment polytope, we prove that the existence of a $(v,w)$-CSCK metric in the first Chern class is equivalent to the existence of a $g(v,w)$-soliton. We also explain that a $g(v,w)$-soliton arises naturally from Sasaki geometry. More precisely, let $(v,w)$ be the weight functions defining a weighted CSCK metric in $2πc_1(X)$ which gives rise to a $\hatξ$-transverse extremal metric on an $S^1$-bundle $N$ in the canonical bundle of a Fano manifold $X$, where $\hatξ$ is a possibly irregular Reeb field on $N$. We prove that the associated $g(v,w)$-soliton on $X$ gives rise to a $\hatξ$-transverse Mabuchi soliton on $N$. |
| title | Soliton-type metrics associated with weighted CSCK metrics on Fano manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2602.06306 |