Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 2 / f / Solution

For $h=z^{-k}f$,The scalar-valued form $h^{-1}dh$ is closed, andBy the period isomorphism from part (b), $[\alpha ]=0$ in complexified de Rham cohomology. Hence there is a smooth complex-valued function $\phi$ on $U_0\cap U_1$ with $\alpha =d\phi$.