3.1. Extension preserving the Lelong class [028I]

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3.1. Extension preserving the Lelong class

Consider the standard embedding

z\in{\mathbb{C}}^{n}\hookrightarrow[1:z]\in{\mathbb{P}}^{n},

where $[t:z]$ denote the homogeneous coordinates on ${\mathbb{P}}^{n}$ . Let $\omega$ be the Fubini-Study Kähler form and let

\rho(t,z)=\log\sqrt{|t|^{2}+\|z\|^{2}}

be its logarithmically homogeneous potential on ${\mathbb{C}}^{n+1}$ .

We denote by $\overline{X}$ the closure of $X$ in ${\mathbb{P}}^{n}$ , so $\overline{X}$ is an algebraic subvariety of ${\mathbb{P}}^{n}$ . It is well known that the class $PSH({\mathbb{P}}^{n},\omega)$ is in one-to-one correspondence with the Lelong class ${\mathcal{L}}({\mathbb{C}}^{n})$ (see [GZ]). Let us look at the connection between $\omega$ -psh functions on $\overline{X}$ and the class ${\mathcal{L}}(X)$ .

The mapping

F_{X}:PSH(\overline{X},\omega\,|_{{}_{\overline{X}}})\longmapsto{\mathcal{L}}(X),\;(F_{X}\varphi)(z)=\rho(1,z)+\varphi([1:z]),

is well defined and injective. However, it is in general not surjective, as shown by Examples 3.2 and 3.3 that follow.

Conversely, a function $\eta\in{\mathcal{L}}(X)$ induces an upper semicontinuous function $\widetilde{\eta}$ on $\overline{X}$ defined in the obvious way:

\widetilde{\eta}([t:z])=\left\{\begin{array}[]{ll}\eta(z)-\rho(1,z),\;\hskip 68.2866pt{\rm if}\;t=1,\;z\in X,\\ \\ \displaystyle\limsup_{[1:\zeta]\to[0:z],\zeta\in X}(\eta(\zeta)-\rho(1,\zeta)),\;{\rm if}\;t=0,\;[0:z]\in\overline{X}\setminus X.\end{array}\right.

The function $\widetilde{\eta}$ is in general only weakly $\omega$ -psh on $\overline{X}$ , i.e. it is bounded above on $\overline{X}$ and it is $\omega\,|_{{}_{\overline{X}_{r}}}$ -psh on the set $\overline{X}_{r}$ of regular points of $\overline{X}$ . This notion is in direct analogy to that of weakly psh function on an analytic variety (see [D2, section 1]). We do not pursue it any further here.

Note that $\eta\in F_{X}\left(PSH(\overline{X},\omega\,|_{{}_{\overline{X}}})\right)$ if and only if $\widetilde{\eta}\in PSH(\overline{X},\omega\,|_{{}_{\overline{X}}})$ . The following simple characterization is a consequence of Theorem B.

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 ${\mathcal{L}}(X)={\mathcal{L}}({\mathbb{C}}^{n})\,|_{{}_{X}}$ .

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]. ∎