The capacity $Cap_{\omega}(\cdot)$ does not distinguish between ”big sets”. Assume indeed there exists an ample divisor $D$ such that $[D]\sim k\omega$, $k\in\mathbb{N}$. Then there exists $\varphi\in PSH(X,\omega)$ such that $dd^{c}\varphi=k^{-1}[D]-\omega$. Note that $\varphi\in{\mathcal{C}}^{\infty}(X\setminus D)$, $e^{\varphi}\in{\mathcal{C}}^{0}(X)$ and $\{\varphi=-\infty\}=D$. Replacing $\varphi$ by $\varphi-\sup_{X}\varphi$ if necessary, we may assume $\sup_{X}\varphi=0$. Consider $\varphi_{c}=\max(\varphi,-c)\in PSH(X,\omega)\cap{\mathcal{C}}^{0}(X)$. Then $\varphi_{c}\equiv\varphi$ outside some neighborhood $V_{c}=\{\varphi<-c\}$ of $D$. Since $0\leq 1+\varphi_{1}\leq 1$ and $\omega_{1+\varphi_{1}}=\omega_{\varphi}=0$ in $X\setminus V_{1}$, we get

hence $Cap_{\omega}(V_{1})=Cap_{\omega}(X)$.

As a concrete example take $X=\mathbb{C}\mathbb{P}^{n}$ and $\omega=\omega_{FS}$, $D$ being some hyperplane $H_{\infty}$ ”at infinity” ($k=1$). Set $\varphi[z:t]=\log|t|-\frac{1}{2}\log[||z||^{2}+|t|^{2}]$ where $z$ denotes the euclidean coordinates in $\mathbb{C}^{n}=\mathbb{C}\mathbb{P}^{n}\setminus H_{\infty}$ and $H_{\infty}=(t=0)$. Observe that $\sup_{\mathbb{C}\mathbb{P}^{n}}\varphi=0$. One then computes

Thus the capacity of the complement of any euclidean ball of radius smaller than $\sqrt{e^{2}-1}$ equals $1$.