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

A connection on a vector bundle is a complex-linear map $D:{\Omega }^0(E)\to {\Omega }^1(E)$ satisfying the Leibniz rule $D(fs)=df\otimes s+fDs$. It is compatible with the Hermitian metric on a holomorphic vector bundle $h$ whenand compatible with the holomorphic structure when its type-$(0,1)$ part is the Dolbeault partial connection: