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Compact quotients of large domains in complex projective space

Finnur Lárusson — 1998

Annales de l'institut Fourier

We study compact complex manifolds covered by a domain in n -dimensional projective space whose complement E is non-empty with ( 2 n - 2 ) -dimensional Hausdorff measure zero. Such manifolds only exist for n 3 . They do not belong to the class 𝒞 , 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...

Convexity of sublevel sets of plurisubharmonic extremal functions

Finnur LárussonPatrice LassereRagnar Sigurdsson — 1998

Annales Polonici Mathematici

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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