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arXiv · 2501.16669

Log truncated threshold and zero mass conjecture

Fusheng Deng·Yinji Li·Qunhuan Liu·Zhiwei Wang·Xiangyu Zhou
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Abstract

For plurisubharmonic functions $φ$ and $ψ$ lying in the Cegrell class of $\mathbb{B}^n$ and $\mathbb{B}^m$ respectively such that the Lelong number of $φ$ at the origin vanishes, we show that the mass of the origin with respect to the measure $(dd^c\max\{φ(z), ψ(Az)\})^n$ 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 $φ$ near the origin in $\mathbb{C}^n$, we introduce a new concept coined the log truncated threshold of $φ$ at $0$ which reflects a singular property of $φ$ via a log function near the origin (denoted by $lt(φ,0)$) and derive an optimal estimate of the residual Monge-Ampère mass of $φ$ at $0$ in terms of its higher order Lelong numbers $ν_j(φ)$ at $0$ for $1\leq j\leq n-1$, in the case that $lt(φ,0)<\infty$. 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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Fusheng Deng, Yinji Li, Qunhuan Liu, Zhiwei Wang, Xiangyu Zhou. 2025-01-28. Log truncated threshold and zero mass conjecture. https://arxiv.org/abs/2501.16669

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