Let $V=\mathbb{C}^{2}/{\pm 1}$. Let $(x,y)$ be the usual affine coordinates on $\mathbb{C}^{2}$, $(u,v,w)$ those on $\mathbb{C}^{3}$. The formulas $u=x^{2},\ v=y^{2},\ w=xy$ realize $V$ as the closed subscheme of $\mathbb{C}^{3}$ whose equation is $uv-w^{2}=0$. We have two ‘natural’Kähler metrics on $V$, the first one is smooth with potential $\varphi^{1}=|u|^{2}+|v|^{2}+|w|^{2}$, induced by the euclidean Kähler metric of $\mathbb{C}^{3}$, the second one is the Kähler current whose potential is $\varphi^{2}=|u|+|v|$. On $V^{reg}$ it is the quotient of the euclidean metric restricted to $\mathbb{C}^{2}-\{0\}$. Near $0$, $dd^{c}\varphi^{2}\gg dd^{c}\varphi^{1}$.