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On 4-dimensional Ricci-flat ALE manifolds
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abstract
In this paper, we prove: 1. There is a one-to-one correspondence between: Hermitian non-K\"ahler ALE gravitational instantons $(M,h)$, and Bach-flat K\"ahler 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-K\"ahler 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-K\"ahler is equivalent to being non-trivially conformally K\"ahler.
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Conformal K\"{a}hler rigidity of Einstein four-manifolds
For compact Einstein four-manifolds, the spectral gap α>β in the self-dual Weyl curvature forces β=γ=−α/2, so the metric is conformally Kähler with positive scalar curvature (after at most a double cover).
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