On Lelong numbers of plurisubharmonic functions on complex spaces
Le Mau Hai, Pham Hoang Hiep, Trinh Tung
TL;DR
This work extends pluripotential theory to singular complex spaces by introducing strong locally irreducible complex spaces and two auxiliary psh functions $\varphi_{aver}$ and $\varphi_{max}$ obtained via ramified coverings and the local parametrization theorem. The main contributions are sharp equalities linking the projective mass $\bar{\nu}_{\varphi}$ to the classical Lelong number $\nu_{\varphi}$ (and to multiplicities) on these spaces, and a Siu-type result showing that upper-level sets of $\nu_{\varphi}$ have analytic structure. The results demonstrate that, under strong local irreducibility, the singularities behave predictably: $\bar{\nu}_{\varphi}(a)=\mathrm{mult}(\widetilde{A},a)\,\nu_{\varphi}(a)$, providing precise control of Lelong numbers in terms of multiplicities. Altogether, the paper supplies new tools and sharp formulas for plurisubharmonic functions on complex spaces, with implications for the study of singularities in several complex variables and complex geometry.
Abstract
In this paper, we introduce the notion of strong locally irreducible complex spaces $\widetilde{X}$. Based on this notion we prove the equality $\barν_{\varphi}(x)=$ mult$(\widetilde{X},x). ν_{\varphi}(x)$ for all $x\in \widetilde{X}$, where $\barν_{\varphi}(x)$ 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 $ν_{\varphi}(x)$ is Lelong number of $\varphi$ at $x$. Moreover, we show that the closure of the upper-level sets $\{z\in \widetilde{X}:ν_{\varphi}(z)\geq c\}$ of a plurisubharmonic function $\varphi$ on a strong locally irreducible complex space $\widetilde{X}$ is a subvariety of $\widetilde{X}$ for all $c\geq 0$.