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Start with any unitary connection ${\nabla }_0$. The Hodge decomposition theorem gives a unique real harmonic differential form $\eta$ representing $[i{F}_{{\nabla }_0}]$. Since they have the same de Rham cohomology class, $\eta -i{F}_{{\nabla }_0}=d\beta$ for a smooth real one-form $\beta$. Set $\nabla ={\nabla }_0-i\beta$. The vector-bundle curvature difference formula givesThis proves existence of a harmonic-curvature unitary line connection for any metric and any complex line bundle.

Two such unitary connections have the same vector-bundle curvature because harmonic representatives are unique. Their difference is therefore $i\gamma$ with $d\gamma =0$. When $b_1(X)=0$, write $\gamma =d\phi$ globally. The bundle gauge transformation $u=e^{i\phi }$ changes $\nabla$ to $u^{-1}\nabla u=\nabla +id\phi$. Thus the harmonic-curvature connection is unique up to gauge. On a general base, closed forms modulo the integral-period forms coming from circle-valued gauge transformations leave the torus $H^1(X;\mathbb{R})/(2\pi H^1(X;\mathbb{Z}))$ of possible gauge classes; the hypothesis removes this freedom.