By Jean-Pierre Demailly

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

**Example text**

Geometric Characterizations of Pseudoconvex Open Sets We first discuss some characterizations of pseudoconvex open sets in Cn . We will need the following elementary criterion for plurisubharmonicity. 1) Criterion. Let v : Ω −→ [−∞, +∞[ be an upper semicontinuous function. Then v is plurisubharmonic if and only if for every closed disk ∆ = z0 + D(1)η ⊂ Ω and every polynomial P ∈ C[t] such that v(z0 + tη) Re P (t) for |t| = 1, then v(z0 ) Re P (0). Proof. The condition is necessary because t −→ v(z0 + tη) − Re P (t) is subharmonic in a neighborhood of D(1), so it satisfies the maximum principle on D(1) by Th.

All properties are immediate consequences of the definition, except perhaps e). That Mη (u1 , . . , up ) is plurisubharmonic follows from a) and Th. 6. Fix a point z0 and ε > 0. All functions u′j (z) = uj (z) − γz0 (z − z0 ) + ε|z − z0 |2 are plurisubharmonic near z0 . It follows that Mη (u′1 , . . , u′p ) = u − γz0 (z − z0 ) + ε|z − z0 |2 is also plurisubharmonic near z0 . Since ε > 0 was arbitrary, e) follows. 19) Corollary. Let uα ∈ ∞ (Ωα ) ∩ Psh(Ωα ) where Ωα ⊂⊂ X is a locally finite open covering of X.

B. Kiselman’s Minimum Principle We already know that a maximum of plurisubharmonic functions is plurisubharmonic. However, if v is a plurisubharmonic function on X × Cn , the partial minimum function on X defined by u(ζ) = inf z∈Ω v(ζ, z) need not be plurisubharmonic. A simple counterexample in C × C is given by v(ζ, z) = |z|2 + 2 Re(zζ) = |z + ζ|2 − |ζ|2 , u(ζ) = −|ζ|2 . It follows that the image F (Ω) of a pseudoconvex open set Ω by a holomorphic map F need not be pseudoconvex. In fact, if Ω = {(t, ζ, z) ∈ C3 ; log |t| + v(ζ, z) < 0} and if Ω′ ⊂ C2 is the image of Ω by the projection map (t, ζ, z) −→ (t, ζ), then Ω′ = {(t, ζ) ∈ C2 ; log |t| + u(ζ) < 0} is not pseudoconvex.