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A sheaf of modules over the sheaf of smooth functions is fine when every locally finite open cover admits a subordinate family of sheaf endomorphisms summing to the identity. Fine sheaves on a paracompact space are acyclic, so their positive-degree sheaf cohomology vanishes.

Order the index set of the open cover $\mathcal{U}=(U_i)$. For the sheaf of abelian groups $\mathcal{F}$, the Čech cochain group isIt is a product, not a finite-support sum. The Čech coboundary isEvery term in ${\delta }^{2}c$ occurs twice with opposite signs, so ${\delta }^2=0$. The Čech cohomology of the cover is ${\check{H}}^p(\mathcal{U},\mathcal{F})=\ker \delta /\mathrm{im}\delta$. Refinements induce maps on these groups; different choices of refinement indices give maps related by a cochain homotopy, hence the same map on cohomology. Taking the direct limit over covers givesOne may use locally finite covers, since a complex manifold is a paracompact space and these covers are cofinal. In degree zero this recovers the group of global sections by the sheaf gluing axiom.

Let ${\mathcal{C}}_{\mathbb{C}}^{\infty }$ be the additive sheaf of complex-valued smooth functions, and ${\mathcal{C}}_{\mathbb{C}}^{\infty ,\ast }$ its multiplicative sheaf of nowhere-zero functions. The smooth exponential sequence isIts kernel is the integer-valued locally constant functions, namely the integer constant sheaf. It is surjective on stalks because a nowhere-zero smooth function has a smooth logarithm on a sufficiently small neighbourhood.

The additive smooth-function sheaf is a fine sheaf, using a partition of unity, and the smooth manifold is a paracompact space. Its positive-degree sheaf cohomology vanishes. The long exact sequence in sheaf cohomology consequently gives an isomorphism

To connect this with bundles, trivialize a smooth complex line bundle over an open cover. Its transition functions $g_{ij}$ form a multiplicative Čech cocycle, satisfying $g_{ij}{g}_{jk}=g_{ik}$. Changing trivializations changes it by a Čech coboundary. Conversely, any such cocycle glues trivial line bundles, and cohomologous cocycles give isomorphic bundles. Thus the isomorphism classes are $H^1(X,{\mathcal{C}}_{\mathbb{C}}^{\infty ,\ast })$.

The connecting isomorphism is the First Chern class. Locally choose logarithms $g_{ij}=e^{2\pi i{f}_{ij}}$; on triple intersectionsrepresents that class. Combining the gluing classification with $\delta$ gives the requested bijection: