Géométrie complexe I

Complex Banach spaces E are existence spaces for classes of plurisubharmonic functions having classes of control sets in E. Two classes F 1 and F 2 are considered here, both are important for applications to complex analysis : F 1 = PSH - ( G ) is the class of the strictly negativ psh-functions in a bounded domain G of E ; F 2 denoted by \mathcal{L}_\sigma(E) \subset P S H(E) is the class of the functions f ( x ) of logarithmic growth σ in E , 0 < σ < +∞. Both F 1 and F 2 have for control sets the not F-pluripolar sets A, giving an upper semi-continous control f(x) \leq \phi_A^*(x, m) for f ∈ F, if f ( x ) ≤ m for x ∈ A. Sets ω in E such B ( O,r ) ∩ ω is B ( O, r ) -pluripolar for r > r 0 are \mathcal{L} -pluripolar. Pluripolar cones in E and sets f ( x ) = — ∞ for f ∈ PSH ( E ) upper bounded in each ball in E must be \mathcal{L} -pluripolar. Sequences of f_n(x) \in \mathcal{L}_\sigma ( E ) with f n ( x ) ↘ —∞ in E are of two different kinds : a ) the convergence is uniform on each bounded set in E, or b ) it is uniform only on \mathcal{L} -pluripolar sets in E . Bounds are obtained for the Lelong number ν ( f,x ) for f ∈ F 1 and x ∈ G' ⊂ G. If W is an analytic set in G of pure codimension p, with the set of regular points W' a connected manifold, then ν ( f,x ) has a constant value ν ( f,W' ) ≥ 0 for x ∈ W', with possible exception for x in a locally pluripolar set A ⊂ W', and ν ( f,x ) > ν ( f,W' ) for x ∈ A. For f ∈ PSH ( G\W ) and W a singularity for f, then p = 1, and a negativ number ν ( f,W' ) is defined; W is a logarithmic singularity if ν ( f,W' ) is finite ; ν ( f,x ) = ν ( f,W' ) holds with possible exception for such a set A ⊂ W' and ν ( f,x ) > ν ( f,W' ) for x ∈ A .