In this question I asked about proving that a connection form $\alpha$ on a $\mathbb{C}^*$ bundle had to have $2\pi i(\alpha  \overline{\alpha})$ be exact. From the answer to that question I understand why on any local trivialization, this quantity will be exact. However, this would ordinarily only imply global exactness if the 1st cohomology is trivial. I am probably being very dense, but how do we show that the global condition follows?
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So if you're going in the direction "$\alpha$invariant pairing exists$\Rightarrow$ $2\pi i(\alpha\overline{\alpha})$ is exact", my previous answer demonstrates that $2\pi i(\alpha\overline{\alpha}) = dg$ at each point. But the function $g$, which is defined as $g(p) = \ln\langle p,p\rangle$, has a global definition on $L\textrm{zero section}$. You're not trying to patch together several local $g$'s. Does this answer your question? 

