From Kähler Ricci solitons to Calabi-Yau Kähler cones
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912307789955072 |
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| author | Apostolov, Vestislav Lahdili, Abdellah Legendre, Eveline |
| author_facet | Apostolov, Vestislav Lahdili, Abdellah Legendre, Eveline |
| contents | We show that if $X$ is a smooth Fano manifold which caries a Kähler Ricci soliton, then the canonical cone of the product of $X$ with a complex projective space of sufficiently large dimension is a Calabi--Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and Nikagawa. This result is obtained by the openness of the set of weight functions $v$ over the momentum polytope of a given smooth Fano manifold, for which a $v$-soliton exists. We discuss other ramifications of this approach, including a Licherowicz type obstruction to the existence of a Kähler Ricci soliton and a Fujita type volume bound for the existence of a $v$-soliton. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02564 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From Kähler Ricci solitons to Calabi-Yau Kähler cones Apostolov, Vestislav Lahdili, Abdellah Legendre, Eveline Differential Geometry Algebraic Geometry 53C55 (primary), 58E11, 53C18, 32Q15, 53C25, 53D10 (secondary) We show that if $X$ is a smooth Fano manifold which caries a Kähler Ricci soliton, then the canonical cone of the product of $X$ with a complex projective space of sufficiently large dimension is a Calabi--Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and Nikagawa. This result is obtained by the openness of the set of weight functions $v$ over the momentum polytope of a given smooth Fano manifold, for which a $v$-soliton exists. We discuss other ramifications of this approach, including a Licherowicz type obstruction to the existence of a Kähler Ricci soliton and a Fujita type volume bound for the existence of a $v$-soliton. |
| title | From Kähler Ricci solitons to Calabi-Yau Kähler cones |
| topic | Differential Geometry Algebraic Geometry 53C55 (primary), 58E11, 53C18, 32Q15, 53C25, 53D10 (secondary) |
| url | https://arxiv.org/abs/2412.02564 |