Plurisubharmonic function

Definition

Let $U\subseteq\mathbb C^d$ be open. A function $u:U\to[-\infty,\infty)$ is plurisubharmonic if it is upper-semicontinuous and, for every affine complex line $L$ , the restriction $u|_{U\cap L}$ is subharmonic or identically $-\infty$ on each component.

Smooth criterion

For $u\in C^2(U)$ , plurisubharmonicity is equivalent to positive semidefiniteness of its Levi form:

\sum_{j,k=1}^d \frac{\partial^2u}{\partial z_j\partial\bar z_k}(z) v_j\overline{v_k}\ge0

for all $z\in U$ and $v\in\mathbb C^d$ .

Comparison with ordinary subharmonicity

Every plurisubharmonic function is subharmonic on the underlying real domain in $\mathbb R^{2d}$ . For a $C^2$ function, the ordinary Laplacian is four times the trace of the Levi matrix, so positivity of the whole Levi form implies positivity of its trace. The converse fails when $d>1$ : ordinary subharmonicity controls only that trace.

The exact relationship with harmonic and pluriharmonic functions is recorded in the H/SH/PSH/PH comparison theorem.

Holomorphic logarithms

If $F:U\to\mathbb C$ is holomorphic and not identically zero, then $\log|F|$ is plurisubharmonic. This makes plurisubharmonic functions the natural potential-theoretic models for magnitudes of entire functions of several variables.

References

Lars Hörmander, Notions of Convexity, Birkhäuser, 2007. DOI record.