From Calabi's extremal metrics to scalar-flat Kähler cones
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Apostolov, Vestislav Lahdili, Abdellah Pan, Chung-Ming |
| author_facet | Apostolov, Vestislav Lahdili, Abdellah Pan, Chung-Ming |
| contents | We prove that for any smooth polarized complex $n$-dimensional manifold $(X, L_X)$ which admits an extremal Kähler metric in $c_1(L_X)$, and for any integer $k$ large enough (in terms of a bound depending on $(X, L_X)$), the $(n+k+1)$-dimensional complex cone $\mathcal{Y}:= \overline{(L_X \otimes \mathcal{O}_{\mathbb{P}^k}(1))^{\times}}$ with section $X \times \mathbb{P}^k$ admits a scalar-flat Kähler cone metric. Equivalently, the unweighted Sasaki join of a smooth compact quasi-regular extremal Sasaki manifold with a regular Sasaki sphere $\mathbb{S}^{2k+1}$ of sufficiently large dimension $(2k+1)$ admits a Sasaki metric of constant (positive) scalar curvature. This gives an affirmative answer to an asymptotic version of a question raised by Boyer--Huang--Legendre--Tønnesen-Friedman in arXiv:1906.04827. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29911 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From Calabi's extremal metrics to scalar-flat Kähler cones Apostolov, Vestislav Lahdili, Abdellah Pan, Chung-Ming Differential Geometry Complex Variables We prove that for any smooth polarized complex $n$-dimensional manifold $(X, L_X)$ which admits an extremal Kähler metric in $c_1(L_X)$, and for any integer $k$ large enough (in terms of a bound depending on $(X, L_X)$), the $(n+k+1)$-dimensional complex cone $\mathcal{Y}:= \overline{(L_X \otimes \mathcal{O}_{\mathbb{P}^k}(1))^{\times}}$ with section $X \times \mathbb{P}^k$ admits a scalar-flat Kähler cone metric. Equivalently, the unweighted Sasaki join of a smooth compact quasi-regular extremal Sasaki manifold with a regular Sasaki sphere $\mathbb{S}^{2k+1}$ of sufficiently large dimension $(2k+1)$ admits a Sasaki metric of constant (positive) scalar curvature. This gives an affirmative answer to an asymptotic version of a question raised by Boyer--Huang--Legendre--Tønnesen-Friedman in arXiv:1906.04827. |
| title | From Calabi's extremal metrics to scalar-flat Kähler cones |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2603.29911 |