Complex Analytic and Differential Geometry 2024, 49: Complex Pseudoconvex Open Sets

An upper semicontinuous function v mapped onto the real space without positive infinity is PSH iff for all closed disk z + Dη in its domain and every polynomial P v(z + tη) ≤ Re[P(t)] for |t| = 1, then v(z) ≤ Re[P(0)]. For complex open subset Ω, the following are equivalent:

Ω is weakly/strongly pseudoconvex

Ω has a PSH exhaustion function

A pseudoconvex open set Ω in a product of complex spaces with dimensions p, n with each slice a complex tube,

An open complex subset Ω with all points on the boundary have a neighborhood V so that their intersection is pseudoconvex is itself pseudoconvex. If it has a C2 boundary, then it's pseudoconvex iff the Levi form on its boundary is semipositive at all points. The boundary is (weakly) strongly pseudoconvex if the Levi-form on the boundary is (semipositive) positive definite, and the boundary is Levi flat, if the Levi-form is 0. If a C1-submanifold Y of a complex analytic manifold X has a complex tangent space T at all points on Y, then Y is complex analytic.