Pseudoconvexity

Summary

Pseudoconvexity is the notion of convexity adapted to several complex variables. An open set in $\C^n$ is pseudoconvex if it admits a continuous plurisubharmonic exhaustion function. Such a function tends to $+\infty$ at the boundary and, for unbounded sets, at infinity. Unlike holomorphic convexity, it is a local condition at the boundary, and for smooth boundaries it can be checked with the Levi form. By the solution of the Levi problem, the pseudoconvex domains in $\C^n$ are exactly the domains of holomorphy.