Complex Analytic and Differential Geometry 2024, 47: Plurisubharmonic Functions

A function u: Ω → [-∞, +∞[ on an open subset on n-dimensional complex space is plurisubharmonic if u is upper semicontinuous, and for every complex line L u is subharmonic on Ω∩L. For any decreasing sequence of plurisubharmonic functions in Psh(Ω), the limit is also plurisubharmonic on Ω. If on every connected component of u, u ≠ -∞ and a family of smoothing kernels p, then u * p is infinitely differentiable and plurisubharmonic on Ω. u * p is non-decreasing and tends to u. For sequences of functions in Psh(Ω), a non-decreasing convex function X is plurisubharmonic on Ω. The sum of the sequence, the maximum of the sequence are also plurisubharmonic on Ω. A locally uniformly bounded from above sequence in Psh(Ω), then the regularized upper envelope is plurisubharmonic and equal to the series supremum almost everywhere. If u ≠ -∞ on all connected component of Ω, then Hu(ξ) is a positive measure.

A holomorphic F: X → Y with v in Psh(Y) has v ⚬ F in Psh(X). A complex Ω with components being open subsets of n-dimensional real space, and u in Psh(Ω) depending exclusively on x = Re z, then u is also convex.

A function u is plurharmonic, if u, -u are plurisubharmonic. If the De Rham cohomology group on X is zero, every PSH function on X can be written u = Re f, with f holomorphic on X.

If the sequence u in Psh(Ω) with Ω in X a locally finite open covering of X, then for every index β at all points z in the boundary of Ω, the maximum of the sequence u is Psh(X), and

A continuous u in Psh(X) is strictly PSH on an open subset Ω in X, with Hu ≥ γ for some continuous positive hermitian form γ on Ω. For any continuous function λ > 0 in C⁰(Ω), there is a PSH v in C⁰(X)∩C∞(Ω) with u ≤ v ≤ u + λ, and v = u on X \ Ω, strictly PSH on Ω. Hv ≥ (1 - λ)γ. v can be is chosen strictly PSH on X, if u has the same property.

A set A ⊂ Ω ⊂ ℝᵐ is polar, if for all x in Ω there is a connected neighborhood W of x and u in Sh(W), u ≠ -∞ so that A ∩ W ⊂ {x in W: u(x) = -∞}. A ⊂ Ω is a closed polar set with v in Sh(Ω \ A) with v bounded above in a neighborhood of all points in A, the v has has a unique extension in Sh(Ω). A closed pluripolar set in a complex analytic manifold X, and all functions v in Psh(X \ A) is locally bounded above near A extends uniquely into Psh(X).