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The Bochner-Hartogs dichotomy for weakly 1-complete Kähler manifolds. (English) Zbl 0904.32008
The B-H (i.e. Bochner-Hartogs) property for a complex manifold \(M\) is \(H^1_c(M,{\mathcal O})= 0\), i.e. for every \(C^\infty\) compactly supported \((0,1)\)-form \(\alpha\) with \(\overline\partial\alpha= 0\) on \(M\) there exists a \(C^\infty\) compactly supported function \(\beta\) on \(M\) with \(\overline\partial\beta= \alpha\).
The Bochner-Hartogs Dichotomy (BHD) says: either \(M\) has the B-H property or there exists a proper holomorphic map of \(M\) onto a Riemann surface. For instance, M. Ramachandran showed [Commun. Anal. Geom. 4, No. 3, 333-337 (1996)] that the universal covering \(M\) of a compact Kähler manifold with a non-constant holomorphic function satisfies (BHD).
The authors give more examples of manifolds with (BHD) in their main theorem: if \(M\) is a connected noncompact weakly 1-complete (i.e. \(M\) admits a continuous plurisubharmonic exhaustion function) Kähler manifold with exactly one end, then \(M\) satisfies (BHD).

MSC:
32Q15 Kähler manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
32F10 \(q\)-convexity, \(q\)-concavity
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