Log truncated threshold and zero mass conjecture
Fusheng Deng, Yinji Li, Qunhuan Liu, Zhiwei Wang, Xiangyu Zhou
TL;DR
This work analyzes singularities of plurisubharmonic functions in the Cegrell class by introducing the log truncated threshold $lt(\varphi,0)$ and linking higher Lelong numbers $\nu_j(\varphi)$ to the residual Monge–Ampère mass $\nu_n(\varphi)$. It proves that if $\nu_1(\varphi)=0$, then for almost every linear map $A$, the residual mass $\nu_n(\max\{\varphi, A^*\psi\})$ vanishes, using a dimension-enlargement approach with $\Phi(z,w)=\max\{\varphi(z),\psi(w)\}$ together with Teissier-type inequalities and Crofton formulas. A sharp bound is established: for finite $lt(\varphi,0)=\gamma$, $\nu_n(\varphi)$ is controlled by the first $n-1$ Lelong numbers and $\gamma$, yielding $\nu_n(\varphi)=0$ when $\nu_1(\varphi)=0$ and providing several corollaries for $S^1$-invariant and uniformly directional Lipschitz PSH functions. The results unite and strengthen known zero-mass statements, offer a new quantitative framework, and illuminate the interplay between directional Lelong numbers and residual Monge–Ampère mass in several complex variables.
Abstract
For plurisubharmonic functions $\varphi$ and $ψ$ lying in the Cegrell class of $\mathbb{B}^n$ and $\mathbb{B}^m$ respectively such that the Lelong number of $\varphi$ at the origin vanishes, we show that the mass of the origin with respect to the measure $(dd^c\max\{\varphi(z), ψ(Az)\})^n$ on $\mathbb{C}^n$ is zero for $A\in \mbox{Hom}(\mathbb{C}^n,\mathbb{C}^m)=\mathbb{C}^{nm}$ outside a pluripolar set. For a plurisubharmonic function $\varphi$ near the origin in $\mathbb{C}^n$, we introduce a new concept coined the log truncated threshold of $\varphi$ at $0$ which reflects a singular property of $\varphi$ via a log function near the origin (denoted by $lt(\varphi,0)$) and derive an optimal estimate of the residual Monge-Ampère mass of $\varphi$ at $0$ in terms of its higher order Lelong numbers $ν_j(\varphi)$ at $0$ for $1\leq j\leq n-1$, in the case that $lt(\varphi,0)<\infty$. These results provide a new approach to the zero mass conjecture of Guedj and Rashkovskii, and unify and strengthen well-known results about this conjecture.