Parabolic exhaustions and analytic coverings.
Compact quotients of large domains in complex projective space
We study compact complex manifolds covered by a domain in -dimensional projective space whose complement is non-empty with -dimensional Hausdorff measure zero. Such manifolds only exist for . They do not belong to the class , so they are neither Kähler nor Moishezon, their Kodaira dimension is , their fundamental groups are generalized Kleinian groups, and they are rationally chain connected. We also consider the two main classes of known 3-dimensional examples: Blanchard manifolds, for which...
Siciak-Zahariuta extremal functions and polynomial hulls
We use our disc formula for the Siciak-Zahariuta extremal function to characterize the polynomial hull of a connected compact subset of complex affine space in terms of analytic discs.
The Siciak-Zahariuta extremal function as the envelope of disc functionals
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.
Convexity of sublevel sets of plurisubharmonic extremal functions
Let X be a convex domain in ℂⁿ and let E be a convex subset of X. The relative extremal function for E in X is the supremum of the class of plurisubharmonic functions v ≤ 0 on X with v ≤ -1 on E. We show that if E is either open or compact, then the sublevel sets of are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.
Affine simplices in Oka manifolds.
Survey of Oka theory.
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