Abstract. A sufficient condition for the infinite dimensionality of the Bergman space of a pseudoconvex domain is given. This condition holds on any pseudoconvex domain that has at least one smooth boundary point of finite type in the sense of D'Angelo.

We show that every strictly pseudoconvex domain Ω with smooth boundary in a complex manifold M admits a global defining function, i.e., a smooth plurisubharmonic function ϕ : U → R defined on an open neighbourhood U ⊂ M of Ω such that Ω = {ϕ < 0}, dϕ = 0 on bΩ and ϕ is strictly plurisubharmonic near bΩ. We then introduce the notion of the core c(Ω) of an arbitrary domain Ω ⊂ M as the set of all points where every smooth and bounded from above plurisubharmonic function on Ω fails to be strictly plurisubharmonic.If Ω is not relatively compact in M, then in general c(Ω) is nonempty, even in the case when M is Stein. It is shown that every strictly pseudoconvex domain Ω ⊂ M with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of c(Ω). We then investigate properties of the core. Among other results we prove 1-pseudoconcavity of the core, we show that in general the core does not possess an analytic structure, and we investigate Liouville type properties of the core.2010 Mathematics Subject Classification. Primary 32T15, 32U05; Secondary 32C15.

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For each n ≥ 2 we construct an unbounded closed pseudoconcave complete pluripolar set E in C n which contains no analytic variety of positive dimension (we call it a Wermer type set). We also construct an unbounded strictly pseudoconvex domain Ω in C n and a smooth CR function f on ∂Ω which has a single-valued holomorphic extension exactly to the set Ω \ E.Example. Let f be an entire function in C 2 andwhere C 1 and C 2 are constants and C 1 > 0. For almost all constants C 2 , Ω is an unbounded strictly pseudoconvex domain with smooth boundary in C 2 containing 2010 Mathematics Subject Classification. Primary 32D10, 32V10, 32T15; Secondary 32D20, 32V25.Key words and phrases. Envelopes of holomorphy, CR functions, strictly pseudoconvex domains, analytic structure.* Supported by the project MURST "Geometric Properties of Real and Complex Manifolds". the divisor {f = 0}. We are going to show that E(∂Ω) is one-sheeted, contained in Ω andFix an exhaustion V 1 ⊂⊂ V 2 ⊂⊂ · · · ⊂⊂ ∂Ω of ∂Ω by relatively compact subsets.Intersecting Ω by balls B 2 (0, R k ) ⊂ C 2 centered at the origin, of radius R k , in such a way that V k ⊂⊂ ∂Ω ∩ B 2 (0, R k ) and then smoothing the edges as in [To], we can find strictly pseudoconvex bounded domains Ω k in C 2 such that V k ⊂ ∂Ω k ∩ ∂Ω for every k ∈ N. Let Γ k := ∂Ω k \ V k . Then, in view of Theorem A from [J], one haswhere Γ k A(Ω k ) is the A(Ω k )-hull of Γ k , i.e., the hull of Γ k with respect to the algebra T. Harz:

We extend the authors’ results on existence of global defining functions to a number of different settings. In particular, we relax the assumption on strict pseudoconvexity of the domain to strict -pseudoconvexity and we consider more general situations, where the ambient space is an almost complex manifold or a complex space. We also investigate to what extent the assumption on smoothness of the boundary of the domains under consideration is necessary in our results. Bibliography: 27 titles.