Motivation for the study of the Chern connection

Question

Given a Hermitian metric $H$ over a holomorphic vector bundle $E$ with holomorphic structure $\overline{\partial}$, there exists a unique connection $\nabla$ (named afer Chern) satisying the following conditions:

1) $\nabla$ is a $H$-connection, i.e H is parallel with respect to $\nabla$,

2) $\nabla^{0,1} = \overline{\partial}$

My question is, what motivates all this? Do you know any application of this? I mean not only in other fields, I am interested also in its role in geometry.

Thank you all for your invaluable help!

Answer 1

The Chern connection coincides with the Levi–Civita connection if and only if $h$ is Kähler. ($h$ is the hermitian metric such that $H(X,Y) = h(X,Y) -ih(JX,Y)$)

Answer 2

One important application is twisted Dolbeault-cohomology. Choosing the chern connection on a bundle allows to extend $ \bar \partial $ on the antiholomorphic differential forms to the bundle $ \Lambda ^{0,\bullet} \otimes E $ s.th. the extension $ \bar \partial_E $ satisfies $ \bar \partial_E^2=0 $. So you can define twisted version of dolbeault-cohomology.