Kähler potential (complex geometry, $\omega =i\partial \bar{\partial }\phi$)

A geometric Kähler potential is locally a real smooth function $\phi$ with $\omega =i\partial \bar{\partial }\phi$ for the Kähler form. Its complex Hessian must be positive definite to define a Kähler metric. Conventions sometimes absorb a factor of $1/2$ into $\phi$. Potentials differing by the real part of a holomorphic function determine the same form. This topic concerns ordinary complex geometry; the existing undisambiguated Kähler potential entry concerns superfields.

For $s=|z_1{|}^2+|z_2{|}^2>0$ and $a>0$, a geometric Kähler potential with derivative ${\phi }^{\prime }(s)=\sqrt{s^2+a^2}/(2s)$ defines a Kähler metric on ${\mathbb{C}}^2\setminus \{0\}$. Its tangential and radial eigenvalues are $\sqrt{s^2+a^2}/(2s)$ and $s/(2\sqrt{s^2+a^2})$. Their product is $1/4$, giving the same Riemannian volume form as ${\omega }_0=(i/2)\sum d{z}_j\wedge d{\bar{z}}_j$, although the metric is different. This construction makes preservation of volume compatible with anisotropic stretching.