Weakly 1-complete surfaces with singularities and applications (Q1001936)

The authors show that if \(X\) is an irreducible complex surface which is weakly \(1\)-complete (i.e., for which there exists a continuous plurisubharmonic exhaustion function) and if there exists a nonconstant holomorphic function on it, then \(X\) is holomorphically convex. As an application, they show that if \(X\) is a holomorphically convex surface which is irreducible and locally irreducible, if \(A\) is a complex curve in \(X\) having no compact connected component, then \(X \backslash A\) is holomorphically convex. Several interesting remarks and examples are also presented.