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 $Ω$ .

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Holomorphic pullbacks of C2C2 plurisubharmonic functions are plurisubharmonic

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

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