Explicit complete Ricci-flat metrics and Kähler-Ricci solitons on direct sum bundles
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913897205727232 |
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| author | Cifarelli, Charles |
| author_facet | Cifarelli, Charles |
| contents | Let $B$ be a Kähler-Einstein Fano manifold, and $L \to B$ be a suitable root of the canonical bundle. We give a construction of complete Calabi-Yau metrics and gradient shrinking, steady, and expanding Kähler-Ricci solitons on the total space $M$, ${\rm dim}_{\mathbb{C}} M = n$ of certain vector bundles $E \to B$, composed of direct sums of powers of $L$. We employ the theory of hamiltonian 2-forms [2, 3] as an Ansatz, thus generalizing recent work of the author and Apostolov on $\mathbb{C}^n$ [5], as well as that of Cao, Koiso, Feldman-Ilmanen-Knopf, Futaki-Wang, and Chi Li [10, 26, 23, 24, 30] when $E$ has Calabi symmetry. As a result, we obtain new examples of asymptotically conical Kähler shrinkers, Calabi-Yau metrics with ALF-like volume growth, and steady solitons with volume growth $R^{\frac{4n-2}{3}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23645 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit complete Ricci-flat metrics and Kähler-Ricci solitons on direct sum bundles Cifarelli, Charles Differential Geometry Let $B$ be a Kähler-Einstein Fano manifold, and $L \to B$ be a suitable root of the canonical bundle. We give a construction of complete Calabi-Yau metrics and gradient shrinking, steady, and expanding Kähler-Ricci solitons on the total space $M$, ${\rm dim}_{\mathbb{C}} M = n$ of certain vector bundles $E \to B$, composed of direct sums of powers of $L$. We employ the theory of hamiltonian 2-forms [2, 3] as an Ansatz, thus generalizing recent work of the author and Apostolov on $\mathbb{C}^n$ [5], as well as that of Cao, Koiso, Feldman-Ilmanen-Knopf, Futaki-Wang, and Chi Li [10, 26, 23, 24, 30] when $E$ has Calabi symmetry. As a result, we obtain new examples of asymptotically conical Kähler shrinkers, Calabi-Yau metrics with ALF-like volume growth, and steady solitons with volume growth $R^{\frac{4n-2}{3}}$. |
| title | Explicit complete Ricci-flat metrics and Kähler-Ricci solitons on direct sum bundles |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2410.23645 |