Levi form of a function

Definition

For a real-valued $C^2$ function $u$ on an open subset of $\mathbb C^d$ , the Levi form at $z$ is the Hermitian form

\mathcal L_u(z;v)= \sum_{j,k=1}^d \frac{\partial^2u}{\partial z_j\partial\bar z_k}(z) v_j\overline{v_k}, \qquad v\in\mathbb C^d.

Its matrix is $\bigl(u_{z_j\bar z_k}\bigr)_{j,k}$ .

Real-coordinate formula

Using $\partial_{z_j}=\tfrac12(\partial_{x_j}-i\partial_{y_j})$ and $\partial_{\bar z_j}=\tfrac12(\partial_{x_j}+i\partial_{y_j})$ , the Levi matrix is a particular Hermitian combination of the blocks of the real Hessian.

Positivity

A $C^2$ function is plurisubharmonic exactly when its Levi form is represented by a positive-semidefinite matrix. A uniform lower bound by a positive Hermitian form is strict plurisubharmonicity.

References

Steven G. Krantz, Function Theory of Several Complex Variables, 2nd ed., AMS Chelsea, 2001. Publisher record.