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arXiv · 2604.26696

Scalar-flat K\"ahler surfaces whose Weyl tensor annihilates the Ricci form

Andrzej Derdzinski·Sinhwi Kim·JeongHyeong Park
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Abstract

We conjecture that any scalar-flat K\"ahler 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 K\"ahler 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

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Andrzej Derdzinski, Sinhwi Kim, JeongHyeong Park. 2026-04-29. Scalar-flat K\"ahler surfaces whose Weyl tensor annihilates the Ricci form. https://arxiv.org/abs/2604.26696

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