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On the structure of complete Kähler manifolds with nonnegative curvature near infinity. (English) Zbl 0695.53052

The main purpose of this article is to study the geometric structure at infinity and the topology of a complete noncompact Kähler manifold M whose sectional curvature is nonnegative outside a compact subset (or for the sake of convenience near infinity). More precisely, the fact is established that if M is of complex dimension m and has more than one end, then except on a compact set, M is metrically a product of a complete complex 1-manifold \(\Sigma\) with a compact Kähler manifold N. The manifold \(\Sigma\) has nonnegative curvature near infinity and hence must be conformally equivalent to a Riemann surface with finite punctures. The compact manifold N has nonnegative sectional curvature and hence its universal cover \(\tilde N\) must be isometrically a product of Kähler manifolds which are either i) isometrically \({\mathbb{C}}^ k\), ii) isometrically an irreducible compact Hermitian symmetric space of rank not less than 2, or iii) isometrically a \({\mathbb{C}}{\mathbb{P}}^ k\) with a nonnegatively curved Kähler metric.
The main technique is to utilize the nonconstant positive harmonic functions. The cases when M has more than one end with one of it being a large end and if M has one large end are considered. The first homology group of M is studied and the first Betti number is estimated from above by 2m-1, where m is the complex dimension of M. In addition, if M has positive sectional curvature near infinity and has a large end, then its first homology group must be trivial.
Reviewer: N.Bokan

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
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