# On the construction of principal $S^1$-bundles with prescribed characteristic form

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 !