Local formula for the Chern connection on a vector bundle ($A=H^{-1}\partial H$)

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 hasThe 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.

At any point of a Hermitian holomorphic vector bundle one may choose a holomorphic local frame with $H=I$ and $\partial H=0$. First normalize $H$ by a constant frame change, then prescribe a holomorphic matrix's first jet to cancel $H^{-1}\partial H$. The Chern matrix vanishes there. This frame reduces first-order bundle identities at that point to scalar identities.