I am trying to understand an argument made by Kobayashi (https://projecteuclid.org/download/pdf_1/euclid.tmj/1178245006, page 35) in his construction of a principal $S^1$-bundle with connection $1$-form having a prescribed curvature form. Below $S^1$ is set to be the integers modulo $1$, $S^1 = \mathbb{R} / \mathbb{Z}$. I begin be recalling some facts.

Let $M$ be a manifold. The inclusion $\mathbb{Z} \hookrightarrow \mathbb{R}$ induces a homomorphism of de Rham cohomology groups $$ H_{D}^2(M;\mathbb{Z}) \to H_{D}^2(M;\mathbb{R}), $$ the image of which is denoted by $H_{D}^2(M;\mathbb{Z})_b$. A closed differential $2$-form $\omega$ is called integral if the cohomology class $[\omega]$ of $\omega$ is an element of $H_{D}^2(M;\mathbb{Z})_b$. It can also be characterized by the fact that $$ \int_{\Sigma} \omega \in \mathbb{Z}, $$ for any closed surface $\Sigma$ in $M$.

If $M$ is endowed with a good cover $\{U_i\}_{i \in I}$, then the isomorphism between the de Rham cohomology $H_{D}^2(M;\mathbb{R})$ and Čech cohomology $H_{C}^2(M;\mathbb{R})$ can be constructed as follows: any closed $2$-form $\omega$ restricts to an exact $1$-form $\omega_i$ on $U_i$, i.e $\omega_{|U_i} = d\omega_i$. Furthermore, since $d(\omega_i - \omega_j)_{|U_i \cap U_j} = 0$, one can write $(\omega_i - \omega_j)_{|U_i \cap U_j} = d \omega_{ij}$, for a function $\omega_{ij}$ on $U_j \cap U_j$ such that $\omega_{ij} = - \omega_{ji}$. But then, one has: $$ d(\omega_{ij} + \omega_{jk} + \omega_{ki})_{|U_i \cap U_j \cap U_k} = 0, $$ so that $$ \omega_{ijk} := \omega_{ij|U_i \cap U_j \cap U_k} + \omega_{jk|U_i \cap U_j \cap U_k}+ \omega_{ki| U_i \cap U_j \cap U_k} $$ is constant. The function $f_{\omega} : (i,j,k) \mapsto \omega_{ijk}$ is a Čech cocycle, and the map $\omega \mapsto f_{\omega}$ provides the desired isomorphism.

In particular, it is easy to see that $\omega$ is integral if and only if $f_{\omega}$ is integer valued, and there is an isomorphism $$ H_{D}^2(M;\mathbb{Z})_b \simeq H_{C}^2(M;\mathbb{Z})_b $$

Given an integral $2$-form $\omega$, let $$ k := \inf \{k’ > 0 | \int_{\Sigma} \omega = k’\}. $$ I would like to understand what the relation is between $k$ and the values of the associated integral Čech cocycle $f_{\omega}$. More precisely, in this situation, Kobayashi constructs a principal $S^1$-bundle and a connection $1$-form with curvature $\omega$, by

« considering that the functions $\omega_{ij}$ are $S^1$-valued, and taking them as coordinate functions, which produces a principal $S^1$-bundle ».

From my understanding, considering $\omega_{ij}$ as $S^1$-valued means composing it with the projection $\mathbb{R} \to \mathbb{R} / \mathbb{Z}$. However, this would change the values of the function $f_{\omega}$ on the intersections $U_i \cap U_j \cap U_k$, $i,j,k \in I$.

Question 1: is this composition equivalent to rescaling the integral form $\omega$? How exactly?

Question 2: what does it mean for the integer $k$ defined above?

Thank you all in advance !



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