= Local real potential for a closed (1,1)-form
{title2=$\omega=i\partial\bar\partial f,\quad f\in C^\infty(U,\mathbb R)$}
For a closed real $(1,1)$ form, the <Poincare lemma> gives a real one-form $\eta$ with $d\eta=\omega$. Its $(0,1)$ part is $\bar\partial$-closed, so the <Dolbeault-Poincaré lemma> gives $\eta^{0,1}=\bar\partial\psi$. Reality gives $\eta^{1,0}=\partial\bar\psi$, hence $\omega=i\partial\bar\partial(2\operatorname{Im}\psi)$. Positivity is a separate requirement for this form to define a metric.
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