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Complex manifolds with split tangent bundle. (English) Zbl 1001.32010
Peternell, Thomas (ed.) et al., Complex analysis and algebraic geometry. A volume in memory of Michael Schneider. Berlin: Walter de Gruyter. 61-70 (2000).
Let $$X$$ be a compact complex manifold with holomorphic tangent bundle $$T_{X}$$ and universal covering space $$\widetilde{X}$$. Assume that $$T_{X}$$ splits holomorphically into a direct sum $$T_{X}=\bigoplus_{i\in I} E_i$$ of holomorphic subbundles $$E_i$$. Under which conditions for $$X$$ is it true that $$\widetilde{X}$$ splits holomorphically into a product $$\widetilde{X}=\prod_{i\in I}U_i$$ of complex submanifolds $$U_i$$ such that the given decomposition of $$T_X$$ lifts to the canonical decomposition $$T_{\widetilde X}\simeq\bigoplus_{i\in I}T_{U_{i}}$$? The author conjectures that it should be sufficient to assume that $$X$$ is Kähler and proves that any of the following three conditions is in fact sufficient: $$X$$ admits a Kähler-Einstein metric, $$T_X$$ is a direct sum of holomorphic line bundles of negative degree, $$X$$ is a Kähler surface. The proof uses the classification of compact complex surfaces.
The Kähler condition is important since a Hopf surface has $$\mathbb C^{2}\backslash\{0\}$$ as universal covering space but its tangent bundle splits into a direct sum of holomorphic line bundles.
For related results see C. T. Simpson [J. Am. Math. Soc. 1, 867-918 (1988; Zbl 0669.58008)] and S. Kobayashi and T. Ochiai [Tohoku Math. J., II Ser. 34, 587-629 (1982; Zbl 0508.32007)].
For the entire collection see [Zbl 0933.00031].

##### MSC:
 32J18 Compact complex $$n$$-folds 32J15 Compact complex surfaces 32Q20 Kähler-Einstein manifolds 32Q55 Topological aspects of complex manifolds
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