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Hi! I've always read that on a complex manifold (obviously not kahler), with a given hermitian metric on tangent bundle, the chern connection and the levi civita connection on the underlying real bundle could be different. Please can someone give me an explicit example of this fact?

Thank you in advance

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up vote 11 down vote accepted

You need a non-Kahler complex manifold. Then the Chern connection will have nontrivial torsion. And the torsion corresponds to the non-closed Kahler form of the metric.

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Further to a previous answer, you might like to refer to arXiv:0911.5655 and references therein for some interesting results on Chern connections on non-Kähler (and indeed non-integrable) almost-Hermitian manifolds.

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