As explained in Section 2, a priori these structures depend on the choice of $\Theta$. But we claim that in our current setting $b_{1}(D)=0$, the choice of $\Theta$ will not change the isomorphism class of the Kähler structures. Given two choices $\Theta$ and $\Theta^{\prime}$, then the difference $\Theta^{\prime}-\Theta$ is a closed 1-form on $Q_{T}\setminus P$. Since $P$ has codimension $3$ in $Q$, we know $H^{1}(Q_{T}\setminus P;\mathbb{R})\simeq H^{1}(Q;\mathbb{R})\simeq H^{1}(D;\mathbb{R})$. Hence we can write

(4.38)

$$\Theta^{\prime}-\Theta=df+\beta$$

for a function $f$ on $Q_{T}\setminus P$ and a harmonic 1-form $\beta$ on $D$. So if $b_{1}(D)=0$ then $\beta=0$, and the isomorphism class of the Kähler structure $(\omega,\Omega)$ does not depend on the choice of $\Theta$. In the general case when $b_{1}(D)>0$, up to gauge equivalence, $\Theta$ and $\Theta^{\prime}$ differ by the pull-back of a flat connection on $D$. In Remark 4.8.2 we shall see the geometric meaning of this.