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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
##### Keywords:
foliations by curves; universal covering
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