Comparison principles for local plurisubharmonic potentials on complex manifolds and applications
Abstract
We prove a weighted comparison principle for local plurisubharmonic potentials in Cegrell's class on an arbitrary complex manifold under suitable boundary and gluing conditions. We obtain domination and uniqueness when the manifold admits a global psh function in ℰloc whose Monge-Ampère measure has an absolutely continuous part with positive density almost everywhere. We apply these results to plurisubharmonic functions on domains in ℂ2 whose singularities satisfy local inequalities involving αlog|f| and a Cegrell potential. For such functions, maximality is equivalent to the vanishing of the Monge-Ampère measure of the current remaining after subtraction of the Siu divisorial part. Consequently, maximality is a local property in this class.
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BibTeX
@misc{do2026comparison,
title = {{Comparison principles for local plurisubharmonic potentials on complex manifolds and applications}},
author = {Thai Duong Do and Pham Hoang Hiep},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05948},
eprint = {2610.05948},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05948},
}
RIS
TY - UNPB
TI - Comparison principles for local plurisubharmonic potentials on complex manifolds and applications
AU - Thai Duong Do
AU - Pham Hoang Hiep
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.05948
UR - https://arxiv.org/abs/2610.05948
AB - We prove a weighted comparison principle for local plurisubharmonic potentials in Cegrell's class on an arbitrary complex manifold under suitable boundary and gluing conditions. We obtain domination and uniqueness when the manifold admits a global psh function in ℰ_loc whose Monge-Ampère measure has an absolutely continuous part with positive density almost everywhere. We apply these results to plurisubharmonic functions on domains in ℂ^2 whose singularities satisfy local inequalities involving αlog|f| and a Cegrell potential. For such functions, maximality is equivalent to the vanishing of the Monge-Ampère measure of the current remaining after subtraction of the Siu divisorial part. Consequently, maximality is a local property in this class.
ER -