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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.01609 |
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| _version_ | 1866915034352844800 |
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| author | Li, Mingyang |
| author_facet | Li, Mingyang |
| contents | In this paper, we prove:
1. There is a one-to-one correspondence between: Hermitian non-Kähler ALE gravitational instantons $(M,h)$, and Bach-flat Kähler orbifolds $(\widehat{M},\widehat{g})$ of complex dimension 2 with exactly one orbifold point $q$, such that the scalar curvature $s_{\widehat{g}}$ satisfies $s_{\widehat{g}}(q)=0$ while being positive elsewhere.
2. There is no Hermitian non-Kähler ALE gravitational instanton $(M,h)$ with structure group contained in $SU(2)$, except for the Eguchi-Hanson metric with reversed orientation.
A 4-dimensional Ricci-flat metric being Hermitian non-Kähler is equivalent to being non-trivially conformally Kähler. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_01609 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On 4-dimensional Ricci-flat ALE manifolds Li, Mingyang Differential Geometry In this paper, we prove: 1. There is a one-to-one correspondence between: Hermitian non-Kähler ALE gravitational instantons $(M,h)$, and Bach-flat Kähler orbifolds $(\widehat{M},\widehat{g})$ of complex dimension 2 with exactly one orbifold point $q$, such that the scalar curvature $s_{\widehat{g}}$ satisfies $s_{\widehat{g}}(q)=0$ while being positive elsewhere. 2. There is no Hermitian non-Kähler ALE gravitational instanton $(M,h)$ with structure group contained in $SU(2)$, except for the Eguchi-Hanson metric with reversed orientation. A 4-dimensional Ricci-flat metric being Hermitian non-Kähler is equivalent to being non-trivially conformally Kähler. |
| title | On 4-dimensional Ricci-flat ALE manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2304.01609 |