Comparaison of Chern Connections (and curvatures) of two metrics

Question

Let $E\rightarrow X$ be a holomorphic vector bundle over a complex manifold. Let $h$ and $k$ be two metrics on $E$ such that $$k(\cdot,\cdot)=h(u\cdot,\cdot)$$ for some bundle endomorphism $u$. I am looking for a good reference that relates the curvatures $F_h$ and $F_k$ of the Chern connections.

Answer 1

Let's calculate in local coordinates. Write $u = u^\alpha_\beta\in \operatorname{GL}(n, \mathbb C)$. Then

$$ k_{\alpha\beta} = u_{\alpha}^\gamma h_{\gamma\beta}$$

or just $k = uh$. Since the chern connection with respect to $h$ is given by $h^{-1} \partial h$,

$$k^{-1} \partial k = h^{-1} u^{-1} \partial u h + h^{-1} u^{-1} u \partial h = h^{-1} u^{-1} \partial u h + h^{-1} \partial h.$$

and thus the Curvatures satisfy

$$ F_k - F_h = \bar\partial (h^{-1} u^{-1} \partial u h).$$