TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-28

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Decreasing limits of plurisubharmonic functions

Statement

Let $Ω \subseteq C^{m}$ be a domain and let $u_{1} \geq u_{2} \geq \dots$ be a decreasing sequence of plurisubharmonic functions on $Ω$ . Put $u (z) = \lim_{n \to \infty} u_{n} (z)$ . Then either $u \equiv - \infty$ on a connected component of $Ω$ , or $u$ is plurisubharmonic on $Ω$ .

Facts & Assumptions

Given: A decreasing sequence $(u_{n})$ of plurisubharmonic functions on a domain $Ω \subseteq C^{m}$ .

[L1]

Plurisubharmonicity is the affine-line subharmonicity test together with upper semicontinuity and the componentwise nontriviality condition (Plurisubharmonic functions).

[L2]

A decreasing limit of subharmonic functions is subharmonic or identically $- \infty$ on the connected component (A decreasing limit of plane subharmonic functions is subharmonic or identically -infinity).

Proof

technique · direct

1.1given

A decreasing limit of upper semicontinuous functions is upper semicontinuous, so $u$ is upper semicontinuous on $Ω$ . If $u \equiv - \infty$ on some connected component, the first alternative of the statement holds and there is nothing further to prove.

2.1L1L2step 1.1∎

Fix $a \in Ω$ and $v \neq 0$ . On every connected component of the line domain ${λ : a + λ v \in Ω}$ , each restriction $λ \mapsto u_{n} (a + λ v)$ is subharmonic or identically $- \infty$ by [L1]. Therefore [L2] makes the limit restriction $λ \mapsto u (a + λ v)$ subharmonic or identically $- \infty$ there. Since the componentwise $- \infty$ alternative was excluded in step 1.1, [L1] now shows that $u$ is plurisubharmonic on $Ω$ .

Depends on

Used by

Behnke-Stein: increasing unions of pseudoconvex domains Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jiří Lebl, Tasty Bits of Several Complex Variables, §2.4 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, Theorem 7 (standard reference, not scraped)