Complex Analytic and Differential Geometry 2025, 24 - Non-Bounded Operators on Hilbert Spaces

Let (X, ω) a weakly pseudoconvex Kaehler manifold, L a hermitian line bundle, E a hermitian vector bundle over X, Y an analytic subset of X with Y = σ^-1(0) for some section of E and p the maximal codim of the irreducible components of Y. A holomorphic section f of K_X⊗L defined in the open set U ⊃ Y so that |σ(x)| < 1. If ∫_U|f|²dV < +∞ and

If X is exhausted by X_c = {x∈X; ψ(x) < c}, then X_c \ Y has a complete Kaehler metric for all c ∈ ℝ. The same conclusion holds for X \ Y if (X, ω) is complete and for C ≥ 0: Θ_E ≤_Grif Cω ⊗ ⟨ , ⟩_E on X. An open subset Ω of ℂⁿ and Y an analytic subset of Ω. For a (p, q-1)-form v with L²_loc coefficients and a (p, q)-form w with L¹_loc coefficients with d''v = w on Ω\Y, then d''v = w on Ω. For PSH functions φ, ψ on Ω with ψ finite and continuous, σ a family of holomorphic functions on Ω, set U = {z∈Ω; |σ(z)|² < e^-ψ(z)} and U' = {z∈Ω; |σ(z)|² < e^ψ(z)}. For all ε > 0 and holomorphic function f on U, there is a holomorphic F on Ω with F_↑Y = f_↑Y and

For a weakly pseudoconvex domain Ω ⊂ ℂⁿ and PSH function φ on Ω, ε > 0 and all z₀∈Ω with integrable e^-φ in neighborhoods of z₀ there is a holomorphic function F on Ω with F(z₀) = 1 and

A PSH function φ on a complex manifold X and set A of points in X so that e^-φ is not locally integrable in some point of A, then A is an analytic subset of X.