By Jean-Pierre Demailly

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**Extra resources for Complex Analytic and Differential Geometry (September 2009 draft)**

**Example text**

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 . 19 is satisfied. We define u= u M(ηα ) (uα ) on X on Ω. 18 (b,e). ηα ) (u1 . . uα ) on Ωβ 1 β α and uα = u on the complement. F. Polar and Pluripolar Sets. Polar and pluripolar sets are sets of −∞ poles of subharmonic and plurisubharmonic functions. Although these functions possess a large amount of flexibility, pluripolar sets have some properties which remind their loose relationship with holomorphic functions.

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.

If u ∈ Psh(Ω), then Hu(ξ) = weak lim H(u ⋆ ρε )(ξ) 0. Conversely, Hv 0 implies H(v ⋆ ρε ) = (Hv) ⋆ ρε 0, thus v ⋆ ρε ∈ Psh(Ω), and also ∆v 0, hence (v ⋆ ρε ) is non decreasing in ε and u = limε→0 v ⋆ ρε ∈ Psh(Ω) by Th. 4. 9) Proposition. The convex cone Psh(Ω) ∩ L1loc (Ω) is closed in L1loc (Ω), and it has the property that every bounded subset is relatively compact. B. Relations with Holomorphic Functions In order to get a better geometric insight, we assume more generally that u is a C 2 function on a complex n-dimensional manifold X.