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

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Holomorphic pullbacks of $C^{2}$ plurisubharmonic functions are plurisubharmonic

Statement

Let $Ω \subseteq C^{m}$ and $V \subseteq C^{n}$ be domains, let $F : Ω \to V$ be holomorphic, and let $u \in C^{2} (V, R)$ be plurisubharmonic. Then $u \circ F$ is plurisubharmonic on $Ω$ .

Facts & Assumptions

Given: Domains $Ω \subseteq C^{m}$ and $V \subseteq C^{n}$ , a holomorphic map $F : Ω \to V$ , and a $C^{2}$ plurisubharmonic function $u : V \to R$ .

[L1]

In the $C^{2}$ setting, plurisubharmonicity is equivalent to semipositivity of the Levi form (The C^2 Levi criterion for plurisubharmonicity, The Levi form and strict plurisubharmonicity).

[L2]

Holomorphic maps compose holomorphically and satisfy the chain rule (The composite of holomorphic maps is holomorphic and its complex Jacobian is the product).

Proof

technique · direct

1.1L2given

For $a \in Ω$ and $v \in C^{m}$ , the chain rule [L2] gives $L_{u \circ F} (a; v) = L_{u} (F (a); D F (a) v) .$

2.1L1step 1.1given∎

Because $u$ is $C^{2}$ and plurisubharmonic, [L1] makes the right-hand side of step 1.1 nonnegative for every $a$ and $v$ . Applying [L1] again, $u \circ F$ is plurisubharmonic on $Ω$ .

Depends on

Used by

A smooth psh exhaustion gives Hartogs pseudoconvexity on bounded domains Lemma

Dependency tree · two levels

22 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, §3.2.4 (standard reference, not scraped)

Holomorphic pullbacks of C2C2 plurisubharmonic functions are plurisubharmonic