TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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

1.1

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.

given

2.1

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 $Ω$ .

L1L2step 1.1∎

Depends on

Used by

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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)