The Levi form and strict plurisubharmonicity

Fix an integer $m\ge 1$. In this Definition, use one-based aliases for the canonical coordinates of Complex $m$-space and its real coordinate dictionary: for $1\le j\le m$, the symbols $z_j$ and $v_j$ denote the canonical coordinate and vector component with index $j-1$. The derivatives ${\mathrm{\partial }}_{z_j}$ and ${\mathrm{\partial }}_{{\bar{z}}_j}$ denote the corresponding canonical Wirtinger operators of Wirtinger operators in ${\mathbb{C}}^m$ with index $j-1$; all derivatives below use these aliases.

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

${\mathcal{L}}_u(a;v):={\sum }_{j=1}^m{\sum }_{k=1}^m\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}_j\mathrm{\partial }{\bar{z}}_k}(a)v_j\overline{v_k}\mathrm{.}$

The function $u$ is strictly plurisubharmonic when

${\mathcal{L}}_u(a;v)>0\text{for every }a\in \mathrm{\Omega }\text{ and every }v\mathrm{\ne }0.$

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