P111

In mathematics, a Hermitian connection $\mathrm{\nabla }$, is a connection on a Hermitian vector bundle over a smooth manifold which is compatible with the Hermitian metric. If the base manifold is a complex manifold, and the Hermitian vector bundle admits a holomorphic structure, then there is a canonical Hermitian connection, which is called the Chern connection which satisfies the following conditions

1. Its (0, 1)-part coincides with the Cauchy-Riemann operator associated to the holomorphic structure.
2. Its curvature form is a (1, 1)-form.

In particular, if the base manifold is Kähler and the vector bundle is its tangent bundle, then the Chern connection coincides with the Levi-Civita connection of the associated Riemannian metric

References

• Shiing-Shen Chern, Complex Manifolds Without Potential Theory.

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