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arXiv · 2610.05948

Comparison principles for local plurisubharmonic potentials on complex manifolds and applications

Thai Duong Do·Pham Hoang Hiep
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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 $\mathcal E_{\mathrm{loc}}$ whose Monge-Ampère measure has an absolutely continuous part with positive density almost everywhere. We apply these results to plurisubharmonic functions on domains in $\mathbb C^2$ whose singularities satisfy local inequalities involving $α\log\|f\|$ 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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Thai Duong Do, Pham Hoang Hiep. 2026-10-05. Comparison principles for local plurisubharmonic potentials on complex manifolds and applications. https://arxiv.org/abs/2610.05948

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