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The Levi form and strict plurisubharmonicity

Definition

Let $Ω \subseteq C^{m}$ be open and let $u \in C^{2} (Ω, R)$ . For $a \in Ω$ and $v \in C^{m}$ , the Levi form of $u$ at $a$ in the direction $v$ is

$L_{u} (a; v) : = \sum_{j = 1}^{m} \sum_{k = 1}^{m} \frac{\partial^{2} u}{\partial z_{j} \partial {\overline{z}}_{k}} (a) v_{j} \overline{v_{k}} .$

The function $u$ is strictly plurisubharmonic when

$L_{u} (a; v) > 0 for every a \in Ω and every v \neq 0.$

Remarks

The Levi form is Hermitian in the vector variable. Semipositivity, $L_{u} (a; v) \geq 0$ for all $v$ , is the condition that characterizes ordinary plurisubharmonicity in the $C^{2}$ setting.

Used by

Levi pseudoconvex domains Definition
The unit ball is Levi pseudoconvex Example
Hartogs pseudoconvexity yields a continuous plurisubharmonic exhaustion Theorem
Holomorphic pullbacks of C² plurisubharmonic functions are plurisubharmonic Theorem
The C² Levi criterion for plurisubharmonicity Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Jiří Lebl, Tasty Bits of Several Complex Variables, §2.3 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.3.1 (standard reference, not scraped)