= Local formula for the Chern connection on a vector bundle
{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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