On Lelong numbers of plurisubharmonic functions on complex spaces

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