As we know, a compact Kähler manifold always satisfies the $\partial\bar{\partial}$-lemma (see for example Huybrechts' book 《complex geometry》p128), thus we call it a $\partial\bar{\partial}$-manifold, it seems not all the $\partial\bar{\partial}$-manifolds are Kähler, but I don't have an example, so may someone provide one? The simpler the better.
2026-03-26 09:37:28.1774517848
Non-Kähler $\partial\bar{\partial}$-manifold
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