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Kähler manifolds with split tangent bundle. (English) Zbl 1187.32018
From the introduction: We study in this paper compact Kähler manifolds whose tangent bundle splits as a sum of two or more subbundles. The basic result that we prove is the following theorem.
Theorem 1.1. Let \(M\) be a compact connected Kähler manifold. Suppose that its tangent bundle \(TM\) splits as \(D\oplus L\), where \(D\subset TM\) is a subbundle of codimension one and \(L\subset TM\) is a subbundle of dimension one. Then: 7mm
(i)
If \(D\) is not integrable, then \(L\) is tangent to the fibres of a \(\mathbb P\)-bundle;
(ii)
If \(D\) is integrable, then \(\widetilde M\), the universal covering of \(M\), splits as \(\widetilde N\times E\), where \(E\) is a connected simply connected curve \((\mathbb D, \mathbb C\) or \(\mathbb P\)). This splitting of \(\widetilde M\) is compatible with the splitting of \(TM\), in the sense that \(T\widetilde N\subset T\widetilde M\) is the pull-back of \(D\) and \(TE\subset T\widetilde M\) is the pull-back of \(L\).

This result is the main ingredient in the proof of Theorem 1.2, resp., Theorem 4.1 of the paper.

MSC:
32Q30 Uniformization of complex manifolds
37F75 Dynamical aspects of holomorphic foliations and vector fields
53C12 Foliations (differential geometric aspects)
58A30 Vector distributions (subbundles of the tangent bundles)
32Q15 Kähler manifolds
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