plurisubharmonic

Plurisubharmonic (psh) functions are a fundamental concept in several complex variables and complex potential theory. Let Ω ⊂ C^n be a domain. A function u: Ω → [−∞, ∞) is plurisubharmonic if it is upper semicontinuous, not identically −∞, and for every holomorphic map φ: D → Ω from the unit disk D ⊂ C, the composition u ∘ φ is subharmonic on D.

For C^2 functions, there is an equivalent differential condition: u is psh if the complex Hessian i∂∂̄u is positive semidefinite, i.e., the Levi form is nonnegative on all complex tangent directions. Equivalently, for all z ∈ Ω and all ξ ∈ C^n, ∑ u_{j k̄}(z) ξ_j ξ̄_k ≥ 0.

Examples include u(z) = log|f(z)| for holomorphic f with no zeros on a chosen domain, u(z) = |z|^2, and any harmonic function (such as Re f for holomorphic f). The defining property that u ∘ φ is subharmonic on every complex line makes psh functions natural in studying holomorphic functions and their domains.

Basic properties: the sum of plurisubharmonic functions is plurisubharmonic, and the pointwise maximum of two plurisubharmonic functions is plurisubharmonic. Psh functions are central to the notion of pseudoconvexity: a domain Ω is pseudoconvex if it admits a continuous plurisubharmonic exhaustion function. When u is C^2 and psh, the complex Monge–Ampère operator (dd^c u)^n yields a positive measure, a key tool in complex geometry and dynamics.