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We study properties of a Green function G A with singularities along a complex subspace A of a complex manifold X. It is defined as the largest negative plurisubharmonic function u satisfying locally u ≤ log |ψ|+C, where ψ = (ψ 1 , . . . , ψ m ), ψ 1 , . . . , ψ m are local generators for the ideal sheaf I A of A, and C is a constant depending on the function u and the generators. A motivation for this study is to estimate global bounded functions from the sheaf I A and thus proving a "Schwarz Lemma" for I A .Subject Classification (2000): Primary 32U35. Secondary 32C15, 32C25, 32H02, 32S45, 32U25, 32U40.

We show that a disc functional on a complex manifold has a plurisubharmonic envelope if all its pullbacks by holomorphic submersions from domains of holomorphy in a‰ne space do and it is locally bounded above and upper semicontinuous in a certain weak sense. For naturally defined classes of disc functionals on manifolds, this result reduces a property somewhat stronger than having a plurisubharmonic envelope to the a‰ne case. The proof uses a recent Stein neighbourhood construction of Rosay, who proved the plurisubharmonicity of the Poisson envelope on all manifolds. As a consequence, the Riesz envelope and the Lelong envelope are plurisubharmonic on all manifolds; for the former, we make use of new work of Edigarian. The basic theory of the three main classes of disc functionals is thereby extended to all manifolds.The first-named author was supported in part by the Natural Sciences and Engineering Research Council of Canada.Brought to you by | The University of York Authenticated Download Date | 7/7/15 7:53 AM

Abstract. We introduce a new type of pluricomplex Green function which has a logarithmic pole along a complex subspace A of a complex manifold X. It is the largest negative plurisubharmonic function on X whose Lelong number is at least the Lelong number of log max{|f 1 |, . . . , |f m |}, where f 1 , . . . , f m are local generators for the ideal sheaf of A. The pluricomplex Green function with a single logarithmic pole or a finite number of weighted poles is a very special case of our construction. We give several equivalent definitions of this function and study its properties, including boundary behaviour, continuity, and uniqueness. This is based on and extends our previous work on disc functionals and their envelopes.

Abstract. We establish disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. This function is also known as the pluricomplex Green function with logarithmic growth or a logarithmic pole at infinity. We extend Lempert's formula for this function from the convex case to the connected case.Introduction. The Siciak-Zahariuta extremal function V X of a subset X of complex affine space C n is defined as the supremum of all entire plurisubharmonic functions u of minimal growth with u|X ≤ 0. It is also called the pluricomplex Green function of X with logarithmic growth or with a logarithmic pole at infinity (although this is a bit of a misnomer if X is not bounded). A plurisubharmonic function u on C n is said to have minimal growth (and belong to the class L) if u − log