### Parabolic exhaustions and analytic coverings.

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We study compact complex manifolds covered by a domain in $n$-dimensional projective space whose complement $E$ is non-empty with $(2n-2)$-dimensional Hausdorff measure zero. Such manifolds only exist for $n\ge 3$. They do not belong to the class $\mathcal{C}$, so they are neither Kähler nor Moishezon, their Kodaira dimension is $-\infty $, 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...

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.

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.

Let X be a convex domain in ℂⁿ and let E be a convex subset of X. The relative extremal function ${u}_{E,X}$ 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 ${u}_{E,X}$ are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.

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