Chern connection in a normal holomorphic frame ($H(x)=I,A(x)=0$)

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.