Scalar-flat Kähler surfaces whose Weyl tensor annihilates the Ricci form
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909019348664320 |
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| author | Derdzinski, Andrzej Kim, Sinhwi Park, JeongHyeong |
| author_facet | Derdzinski, Andrzej Kim, Sinhwi Park, JeongHyeong |
| contents | We conjecture that any scalar-flat Kähler surface in which the Weyl tensor acting on 2-forms annihilates the Ricci form must be either Ricci-flat or locally isometric to a Riemannian product of two real surfaces with mutually opposite nonzero constant Gaussian curvatures. This amounts to the nonexistence of proper weakly Einstein anti-self-dual Kähler surfaces. We prove the above conjecture in three special cases: when the manifold is compact, when one of the Ricci eigendistributions is integrable, and when the norms of the Ricci and Weyl tensors are functionally dependent |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26696 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Scalar-flat Kähler surfaces whose Weyl tensor annihilates the Ricci form Derdzinski, Andrzej Kim, Sinhwi Park, JeongHyeong Differential Geometry 53B35, 53C55 We conjecture that any scalar-flat Kähler surface in which the Weyl tensor acting on 2-forms annihilates the Ricci form must be either Ricci-flat or locally isometric to a Riemannian product of two real surfaces with mutually opposite nonzero constant Gaussian curvatures. This amounts to the nonexistence of proper weakly Einstein anti-self-dual Kähler surfaces. We prove the above conjecture in three special cases: when the manifold is compact, when one of the Ricci eigendistributions is integrable, and when the norms of the Ricci and Weyl tensors are functionally dependent |
| title | Scalar-flat Kähler surfaces whose Weyl tensor annihilates the Ricci form |
| topic | Differential Geometry 53B35, 53C55 |
| url | https://arxiv.org/abs/2604.26696 |