[Math] chern connection vs levi-civita connection

Question

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

Answer 1

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.