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If $0\to \mathcal{F}\to {\mathcal{I}}^0\to {\mathcal{I}}^1\to \cdots$ is an exact resolution of sheaves and each ${\mathcal{I}}^j$ has vanishing positive sheaf cohomology, then the cohomology of its global-section complex computes $H^q(X,\mathcal{F})$. Fine sheaves provide acyclic terms on a paracompact manifold, giving the analytic Dolbeault resolution of holomorphic differential forms.

The Dolbeault-Poincaré lemma makes this a locally exact complex of sheaves, with degree-zero kernel the holomorphic differential forms. The sheaves of smooth forms are fine sheaves by a partition of unity. The acyclic resolution theorem therefore computes the cohomology of ${\Omega }^p$ by the global Dolbeault operator complex.

For the sheaf ${\Omega }^p$ of holomorphic $p$-forms on a complex manifold, the Dolbeault-Poincaré lemma makesa resolution by fine sheaves. Taking global sections therefore gives the natural isomorphism

The Dolbeault theorem identifieswhere ${\Omega }_X^p$ is the sheaf of holomorphic differential forms of degree $p$.

The local analytic ingredient is the Dolbeault-Poincaré lemma: a $\bar{\partial }$-closed smooth $(p,q)$-form with $q>0$ is locally $\bar{\partial }$-exact. Here is a local proof. On a polydisc, the one-variable Cauchy-Green operator in coordinate $z_j$ iswith a smooth cutoff supported in the coordinate disc and equal to one on a smaller disc. The fundamental-solution identity ${\partial }_{\bar{z}}(1/(\pi z))={\delta }_0$ gives ${\partial }_{{\bar{z}}_j}{T}_{j}g=g$ on that smaller disc. The operator is smooth in the parameters and commutes with derivatives in the other coordinates.

For a $(0,q)$-form, write $\alpha =d{\bar{z}}_m\wedge \beta +\gamma$, with neither $\beta$ nor $\gamma$ containing $d{\bar{z}}_m$. Subtract $\bar{\partial }(T_m\beta )$. The remainder contains no $d{\bar{z}}_m$, remains closed, and its coefficients are holomorphic in $z_m$. Repeat with $z_{m-1}$ and then the other coordinates, shrinking discs as needed. Each new integral preserves the holomorphic dependence already achieved. At the end the closed remainder has positive antiholomorphic degree but contains no antiholomorphic differential, so it is zero. Factoring out each holomorphic basis form $d{z}_I$ gives the same assertion for $(p,q)$-forms, with the fixed degree sign included in the primitive. At degree zero, the kernel of $\bar{\partial }$ consists precisely of holomorphic coefficients.

Thus the following is an exact Dolbeault resolution of holomorphic differential forms of sheaves:Each sheaf of smooth forms is a fine sheaf: multiplication by a smooth partition of unity supplies endomorphisms supported in its open cover. Fine sheaves on a paracompact manifold have zero positive sheaf cohomology; the Čech contraction sums a cochain against the partition of unity, giving $\delta h+h\delta =I$ in positive degrees. The general acyclic resolution theorem therefore computes ${\check{H}}^q(X,{\Omega }_X^p)$ as the cohomology of the global-section complex above. That complex is exactly the Dolbeault resolution of holomorphic differential forms, proving the claimed isomorphism. The sheaf-cohomology/Čech comparison and the acyclic-resolution principle are the stated general Čech properties used here.

For a holomorphic vector bundle $E$, the sheaf of vector-bundle-valued differential forms iswhere sections are smooth and restrictions are the usual ones. In a holomorphic local frame of $E$, the Dolbeault operator ${\bar{\partial }}_E$ acts coefficientwise and satisfies ${\bar{\partial }}_E^2=0$. Its Dolbeault cohomology with values in a holomorphic vector bundle is the cohomology of the global section complex ${\mathcal{A}}^{p,\bullet }(X,E)$.

The bundle-valued Dolbeault theorem identifies this with sheaf cohomology:Here ${\Omega }_X^p$ is the sheaf of holomorphic differential forms, and $\mathcal{O}(E)$ is the sheaf of holomorphic sections of a vector bundle. Equivalently, the complexis a fine sheaf resolution, by the local Dolbeault-Poincaré lemma and smooth partitions of unity. In particular $p=0$ computes $H^q(X,\mathcal{O}(E))$; for $p>0$ the holomorphic differential-form factor must be retained.

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:

Let ${\Omega }^p$ be the sheaf of holomorphic $p$-forms and ${\mathcal{A}}^{p,q}$ the sheaf of smooth $(p,q)$-forms. The Dolbeault-Poincaré lemma makesan exact resolution by fine sheaves, which are acyclic for global sections. Its global cochain complex computes Dolbeault cohomology, proving the Dolbeault theorem

On a smooth manifold, ${\mathcal{A}}^k(U)$ is the space of smooth complex-valued degree-$k$ differential forms on $U$. Restriction of forms supplies the sheaf maps. Smooth partitions of unity make this a fine sheaf.

On a smooth manifold, the sequence of sheaves of abelian groupsis exact. The additive smooth-function sheaf is a fine sheaf, so the connecting map identifies $H^1(X,{\mathcal{C}}_{\mathbb{C}}^{\infty ,\ast })$ with $H^2(X,\mathbb{Z})$. The former classifies smooth complex line bundles and the connecting class is their First Chern class.