LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Affine reparametrization does not change the line-test definition

Statement

In the definition of plurisubharmonicity, the condition on the restriction to an affine complex line is unchanged if that line is reparametrized by a nonconstant affine map of one complex variable.

Facts & Assumptions

Given: A domain $Ω \subseteq C^{m}$ , a function $u : Ω \to [- \infty, \infty)$ , an affine line map $λ \mapsto a + λ v$ with $v \neq 0$ , and a nonconstant affine change of variable $ϕ (μ) = α μ + β$ with $α \neq 0$ .

[L1]

Plurisubharmonicity is defined by asking the line restriction to be subharmonic or identically $- \infty$ on each connected component (Plurisubharmonic functions).

[L2]

Plane subharmonicity is the upper-semicontinuous disc-submean condition (Subharmonic functions on plane domains).

Proof

technique · direct

1.1

The two parametrizations describe the same affine line because $a + (α μ + β) v = (a + β v) + μ (α v)$ . Thus the second restriction is just $(u_{a, v}) \circ ϕ$ on the corresponding one-variable domain.

L1givenalgebra

2.1

A nonconstant affine map sends discs to discs and is biholomorphic onto its image, so the upper-semicontinuity and disc-submean inequalities of [L2] are preserved under composition with $ϕ$ and with $ϕ^{- 1}$ . Therefore $u_{a, v}$ is subharmonic or identically $- \infty$ on one component exactly when $(u_{a, v}) \circ ϕ$ is subharmonic or identically $- \infty$ on the corresponding component. By [L1], the line-test definition is independent of the chosen affine parametrization.

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, §3.2.4 (standard reference, not scraped)