In a <holomorphic local frame> $e$, let $H=(h(e_i,e_j))$ and write $De=eA$. Holomorphic compatibility forces $A^{0,1}=0$, while <metric compatibility> forces the <local formula for the Chern connection on a vector bundle>
$$
\boxed{A=H^{-1}\partial H.}
$$
This proves uniqueness. The formula transforms by the <connection one-form> law under a holomorphic frame change, so the local definitions glue and satisfy both conditions. This proves existence of the unique <Chern connection>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 4 / c / Solution

{title2=$A=H^{-1}\partial H$}

In a <holomorphic local frame> of a Hermitian <holomorphic vector bundle>, let $H$ be the matrix of its <Hermitian metric on a holomorphic vector bundle>. With the convention $\nabla e=eA$, the <Chern connection> has
$$
A=H^{-1}\partial H.
$$
The condition $\nabla^{0,1}=\bar\partial_E$ forces $A$ to have type $(1,0)$, while <metric compatibility> forces this formula, proving both existence and uniqueness.

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