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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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