Proposition 3.1 . [028J]

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Proposition 3.1.

Let $\eta\in{\mathcal{L}}(X)$ . The following are equivalent:

(i) There exists $\psi\in{\mathcal{L}}({\mathbb{C}}^{n})$ so that $\psi=\eta$ on $X$ .

(ii) $\widetilde{\eta}\in PSH(\overline{X},\omega\,|_{{}_{\overline{X}}})$ .

(iii) For every point $a\in\overline{X}\setminus X$ the following holds: if $(X_{j},a)$ are the irreducible components of the germ $(\overline{X},a)$ then the value

\limsup_{X_{j}\ni[1:\zeta]\to a}(\eta(\zeta)-\rho(1,\zeta))

is independent of $j$ .

In particular, if the germs $(\overline{X},a)$ are irreducible for all points $a\in\overline{X}\setminus X$ then ${\mathcal{L}}(X)={\mathcal{L}}({\mathbb{C}}^{n})\,|_{{}_{X}}$ .