Log truncated threshold and zero mass conjecture

Abstract

For plurisubharmonic functions $\ddot{I}†$ and $\ddot{I}ˆ$ lying in the Cegrell class of ${\mathbb{B}}^n$ and ${\mathbb{B}}^m$ respectively such that the Lelong number of $\ddot{I}†$ at the origin vanishes, we show that the mass of the origin with respect to the measure $(d{d}^c\max \{\ddot{I}†(z),\ddot{I}ˆ(Az)\}{)}^n$ 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 $\ddot{I}†$ near the origin in ${\mathbb{C}}^n$, we introduce a new concept coined the log truncated threshold of $\ddot{I}†$ at $0$ which reflects a singular property of $\ddot{I}†$ via a log function near the origin (denoted by $lt(\ddot{I}†,0)$) and derive an optimal estimate of the residual Monge-Ampère mass of $\ddot{I}†$ at $0$ in terms of its higher order Lelong numbers $\hat{I}{½}_j(\ddot{I}†)$ at $0$ for $1\le j\le n-1$, in the case that $lt(\ddot{I}†,0)<\infty$. 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.

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