By Jean-Pierre Demailly
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B(zj ,εj ) As every ball B(z, ε) is a union of balls B(zj , εj ), we easily conclude that v ⋆ v ⋆ = u⋆ . 23) Choquet’s lemma. Every family (uα ) has a countable subfamily (vj ) = (uα(j) ) such that its upper envelope v satisfies v u u⋆ = v ⋆ . 24) Proposition. If all uα are subharmonic, the upper regularization u⋆ is subharmonic and equal almost everywhere to u. Proof. By Choquet’s lemma we may assume that (uα ) is countable. Then u = sup uα is a Borel function. As each uα satisfies the mean value inequality on every ball B(z, r) ⊂ Ω, we get u(z) = sup uα (z) sup µB (uα ; z, r) µB (u ; z, r).
Every point x ∈ X has a neighborhood V that for any y ∈ V {x} there exists f ∈ Ç(X) with f (y) = f (x). such The second condition is automatic if X = Ω is an open subset of Cn . Hence an open set Ω ⊂ Cn is Stein if and only if Ω is a domain of holomorphy. 17) Lemma. 16 b) of local separation, there exists a smooth nonnegative strictly plurisubharmonic function u ∈ Psh(X). Proof. Fix x0 ∈ X. We first show that there exists a smooth nonnegative function u0 ∈ Psh(X) which is strictly plurisubharmonic on a neighborhood of x0 .
Log rn ), m(u ; r1 , . . , rn ) = m(log r1 , . . , log rn ). 42 Chapter I. D. 15) Definition. A function u is said to be pluriharmonic if u and −u are plurisubharmonic. e. d′ d′′ u = 0 or ∂ 2 u/∂zj ∂z k = 0 for all j, k. If f ∈ Ç(X), it follows that the functions Re f, Im f are pluriharmonic. 16) Theorem. If the De Rham cohomology group HDR (X, R) is zero, every pluriharmonic function u on X can be written u = Re f where f is a holomorphic function on X. 1 Proof. By hypothesis HDR (X, R) = 0, u ∈ ∞ (X) and d(d′ u) = d′′ d′ u = 0, hence there ∞ exists g ∈ (X) such that dg = d′ u.
Complex Analytic and Differential Geometry (September 2009 draft) by Jean-Pierre Demailly
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