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The Bochner-Hartogs dichotomy for bounded geometry hyperbolic Kähler manifolds. (La dichotomie de Bochner-Hartogs pour les variétés kählériennes hyperboliques à géométrie bornée.) (English. French summary) Zbl 1361.32029

Let \(X\) be a noncompact complex manifold and let \(H^1_c(X,\mathcal O)\) be the first compactly supported cohomology with values in the structure sheaf. \(X\) is said to have the Bochner-Hartogs property if \(H^1_c(X,\mathcal O)=0\). This paper is devoted to the study of conditions under which a connected noncompact complete Kähler manifold \(X\) satisfies the following Bochner-Hartogs dichotomy: either \(H^1_c(X,\mathcal O)=0\) or \(X\) admits a proper holomorphic mapping onto a Riemann surface. A connected noncompact oriented Riemannian manifold is called hyperbolic if it admits a positive symmetric Green’s function. The main result of this paper is that a connected noncompact hyperbolic complete Kähler manifold, with bounded geometry of order two and exactly one end, satisfies the Bochner-Hartogs dichotomy.

MSC:

32Q15 Kähler manifolds
32D15 Continuation of analytic objects in several complex variables
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