Proof. [028K]

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Proof.

Assume that $(i)$ holds. It follows that $\widetilde{\eta}=\varphi\,|_{{}_{\overline{X}}}$ , where

\varphi([t:z]):=\left\{\begin{array}[]{ll}\psi(z)-\rho(1,z),\;\hskip 82.51299pt{\rm if}\;t=1,\\ \limsup_{[1:\zeta]\to[0:z]}(\psi(\zeta)-\rho(1,\zeta)),\;{\rm if}\;t=0,\end{array}\right.

is an $\omega$ -psh function on ${\mathbb{P}}^{n}$ . Hence $\widetilde{\eta}\in PSH(\overline{X},\omega\,|_{{}_{\overline{X}}})$ .

Conversely, if $(ii)$ holds then by Theorem B there exists an $\omega$ -psh function $\varphi$ on ${\mathbb{P}}^{n}$ which extends $\widetilde{\eta}$ . Hence $\psi(z)=\rho(1,z)+\varphi([1:z])$ is an extension of $\eta$ and $\psi\in{\mathcal{L}}({\mathbb{C}}^{n})$ .

The equivalence of $(ii)$ and $(iii)$ follows easily from [D2, Theorem 1.10]. ∎