On Lelong numbers of plurisubharmonic functions on complex spaces

Abstract

In this paper, we introduce the notion of strong locally irreducible complex spaces $\widetilde{X}$. Based on this notion we prove the equality $\bar{\nu}_{\varphi}(x)=$ mult$(\widetilde{X},x). \nu_{\varphi}(x)$ for all $x\in \widetilde{X}$, where $\bar{\nu}_{\varphi}(x)$ is the projective mass of a plurisubharmonic function $\varphi$ at $x$ and mult$(\widetilde{X},x)$ is the multiplicity of $\widetilde{X}$ at $x$ and $\nu_{\varphi}(x)$ is Lelong number of $\varphi$ at $x$. Moreover, we show that the closure of the upper-level sets $\{z\in \widetilde{X}:\nu_{\varphi}(z)\geq c\}$ of a plurisubharmonic function $\varphi$ on a strong locally irreducible complex space $\widetilde{X}$ is a subvariety of $\widetilde{X}$ for all $c\geq 0$.