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On the residual Monge–Ampère mass of plurisubharmonic functions with symmetry in \({{\mathbb {C}}}^2\)

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Abstract

The aim of this paper is to study the residual Monge–Ampère mass of a plurisubharmonic function with isolated singularity at the origin in \({{\mathbb {C}}}^2\). We prove that the residual mass is zero if its Lelong number is zero at the origin, provided that it is \(S^1\)-invariant. This result answers the zero mass conjecture raised by Guedj and Rashkovskii in this special case. More generally, we obtain an estimate on the residual mass by the maximal directional Lelong number and Lelong number at the origin.

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Acknowledgements

The author is very grateful to Prof. Xiuxiong Chen and Prof. Mihai Păun for their continuous support and encouragement in mathematics. This problem has been raised to the author when he was studying in Fourier Institute, Grenoble. It is also a great pleasure to thank Chengjian Yao, Xiaojun Wu, Jian Wang, Wei Sun for lots of useful discussions. Finally, the author wishes to thank Prof. Berndtsson, Prof. Rashkovskii, Prof. Xiangyu Zhou and Prof. Chi Li for their valuable suggestions on the first version of this paper.

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  1. Institute of Mathematical Science at ShanghaiTech University, 393 Middle Huaxia Road, Pudong, Shanghai, 201210, China

    Long Li

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Li, L. On the residual Monge–Ampère mass of plurisubharmonic functions with symmetry in \({{\mathbb {C}}}^2\). Math. Z. 306, 13 (2024). https://doi.org/10.1007/s00209-023-03404-5

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