# Complex analytic and differential geometry by Demailly J.-P. By Demailly J.-P.

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Then, for every point y ∈ ∂B, there exists a holomorphic function f ∈ Ç(X) such that f (y) = f (x0). Replacing f with λ(f − f (x0)), we can achieve f (x0) = 0 and |f (y)| > 1. By compactness of ∂B, we find finitely many functions f1 , . . , fN ∈ Ç(X) such that v0 = |fj |2 satisfies v0 (x0 ) = 0, while v0 1 on ∂B. Now, we set u0 (z) = v0 (z) on X 2 Mε {v0 (z), (|z| + 1)/3} on B. 18. Then u0 is smooth and plurisubharmonic, coincides with v0 near ∂B and with (|z|2 + 1)/3 on a neighborhood of x0 .

Let (Ωα ) be a locally finite open covering of Ω by relatively compact open balls contained in coordinate patches of X. Choose concentric balls Ω′′α ⊂ Ω′α ⊂ Ωα of respective radii rα′′ < rα′ < rα and center z = 0 in the given coordinates z = (z1 , . . , zn ) near Ωα , such that Ω′′α still cover Ω. We set uα (z) = u ⋆ ρεα (z) + δα (rα′2 − |z|2 ) on Ωα . For εα < εα,0 and δα < δα,0 small enough, we have uα u + λ/2 and Huα on Ωα . Set ηα = δα min{rα′2 − rα′′2 , (rα2 − rα′2 )/2}. (1 − λ)γ Choose first δα < δα,0 such that ηα < minΩα λ/2, and then εα < εα,0 so small that ′′ u u ⋆ ρεα < u + ηα on Ωα .

18) Lemma. For arbitrary η = (η1 , . . , ηp ) ∈ ]0, +∞[p, the function Mη (t1 , . . , tp ) = Rn max{t1 + h1 , . . , tp + hp } θ(hj /ηj ) dh1 . . dhp 1 j n possesses the following properties: a) Mη (t1 , . . , tp ) is non decreasing in all variables, smooth and convex on Rn ; b) max{t1 , . . , tp } Mη (t1 , . . , tp ) max{t1 + η1 , . . , tp + ηp } ; c) Mη (t1 , . . ,ηp ) (t1 , . . , tj , , . . , tp ) if tj + ηj maxk=j {tk − ηk } ; d) Mη (t1 + a, . . , tp + a) = Mη (t1 , . . , tp ) + a, ∀a ∈ R ; e) if u1 , .