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

In a unitary frame the metric matrix is $I$. If $A$ is the connection one-form, metric compatibility gives $0=dI=A^{\ast }+A$, sothe matrix is Skew-Hermitian. In a holomorphic local frame, every frame vector is annihilated by ${\bar{\partial }}_E$, and compatibility with the holomorphic structure gives