In a unitary frame the metric matrix is $I$. If $A$ is the <connection one-form>, <metric compatibility> gives $0=dI=A^*+A$, so
$$
\boxed{A^*=-A;}
$$
the 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
$$
\boxed{A^{0,1}=0.}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-118/4/b/solution