On Lelong-Bremermann Lemma

Abstract:

The main theorem of this note is the following refinement of the well-known Lelong-Bremermann Lemma: Let $u$ be a continuous plurisubharmonic function on a Stein manifold $\Omega$ of dimension $n.$ Then there exists an integer $m\leq 2n+1$, natural numbers $p_{s}$, and analytic mappings $G_{s}=\left ( g_{j}^{\left ( s\right ) }\right ) :\Omega \rightarrow \mathbb {C}^{m},\ s=1,2,...,$ such that the sequence of functions \begin{equation*} u_{s}\left ( z\right ) =\frac {1}{p_{s}}\max \left ( \ln \left \vert g_{j}^{\left ( s\right ) }\left ( z\right ) \right \vert :\text { }j=1,\ldots ,m\right ) \end{equation*} converges to $u$ uniformly on each compact subset of $\Omega$. In the case when $\Omega$ is a domain in the complex plane, it is shown that one can take $m=2$ in the theorem above (Section 3); on the other hand, for $n$-circular plurisubharmonic functions in $\mathbb {C}^{n}$ the statement of this theorem is true with $m=n+1$ (Section 4). The last section contains some remarks and open questions.