Minimal kernels of weakly complete spaces

Weakly complete compex surfaces

“…(c) If dim C X = 2 and c is a regular value of ϕ, then the (nonempty) level sets Σ 1 c are compact sets foliated by Riemann surfaces (cf. Lemma 4.1 in [23]). If X is a weakly complete complex surface, it is possible to show that (b) and (c) hold for any plurisubharmonic exhaustion function and not just for the minimal ones (see Theorem 3.2).…”

“…This set was introduced in [23] for any weakly complete complex space; let us recall its definition. Given any plurisubharmonic exhaustion function ϕ ∈ C ∞ (X), let Σ 1 ϕ be the minimal closed set such that ϕ is strictly plurisubharmonic on X \ Σ 1 ϕ , and set Σ 1 = Σ 1 (X) = ϕ Σ 1 ϕ (i.e., x ∈ Σ 1 if no plurisubharmonic C ∞ exhaustion function is strictly plurisubharmonic near x).…”

“…A plurisubharmonic exhaustion function ϕ is called minimal if Σ 1 = Σ 1 ϕ . The following crucial properties were proved in [23] when X is a complex manifold:…”

“…[23,Theorem 3.6]). (c) If dim C X = 2 and c is a regular value of ϕ, then the (nonempty) level sets Σ 1 c are compact sets foliated by Riemann surfaces (cf.…”

“…Then, by Theorem 3.2, Y has the local maximum property, and so Theorem 3.9 of [23] applies: there is an s < c such that, if K is the connected component of the set {s ≤ α ≤ c} containing Y , then its topological boundary bK is contained in {α = s} ∪ K, and furthermoreK is nonempty. Then, the forms…”