A note on small deformations of balanced manifolds (Q639600)

Let \((X,J)\) be a compact complex \(n\)-dimensional manifold. Let \(g\) be a Hermitian metric on \(X\) and \(\omega\) be the Hermitian form associated to \(g\). If \(d\omega^{n-1}=0\), then \(g\) or \(\omega\) is called a balanced metric. The authors prove that, under a weaker condition than the \(\partial\bar\partial\)-lemma, the existence of balanced metrics is preserved under small deformations. The weaker condition is satisfied on the twistor space \(Z\) over a compact self-dual four-manifold with Dolbeault cohomology group \(H^{2.0}(Z,\mathbb C)=0\).