With respect to the choice of the connection form $\Theta$ given in (4.67), $(\omega,\Omega)$ defined by (4.34) and (4.35) gives a $C^{2,\alpha}$ (for all $\alpha\in(0,1)$) Kähler structure on $\mathbb{L}$ which is invariant under the natural $S^{1}$-action and is smooth outside $\mathcal{P}$.