Local real potential for a closed (1,1)-form ($\omega =i\partial \bar{\partial }f,f\in C^{\infty }(U,\mathbb{R})$)

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 $\omega =i\partial \bar{\partial }(2\mathrm{Im}\psi )$. Positivity is a separate requirement for this form to define a metric.

A smooth real rotation-invariant potential is constant on every positive-radius circle, so its logarithmic-radius function $u$ is smooth on $\mathbb{R}$. Off the origin, $f_{z\bar{z}}=e^{-t}{u}^{\prime \prime }(t)$. Since $f$ is smooth on the entire plane, this coefficient extends smoothly through the origin. A finite Taylor expansion, rather than a convergent series assumption, gives $f(z)=f(0)+b|z{|}^2+O(|z{|}^4)$.

For a potential already smooth on all of $\mathbb{C}$, the associated form is a Kähler form exactly when the displayed two positivity conditions hold. The first gives a positive metric coefficient off the origin; the second is positivity of $f_{z\bar{z}}(0)$. The associated real metric is $2f_{z\bar{z}}(d{x}^2+d{y}^2)$. Positivity away from the origin alone permits a degenerate metric at the origin.