Dolbeault isomorphism

Remarks

Let $\mathcal{A}^{p,q}$ be the sheaf of smooth $(p,q)$-forms. The Dolbeault–Grothendieck lemma shows that the sequence of sheaves $$0 \to \Omega^p_X \to \mathcal{A}^{p,0} \xrightarrow{\bar\partial} \mathcal{A}^{p,1} \xrightarrow{\bar\partial} \mathcal{A}^{p,2} \to \cdots$$ is exact. The sheaves $\mathcal{A}^{p,q}$ are fine, because they are modules over the sheaf of smooth functions, which has partitions of unity, so they have no higher cohomology. The cohomology of $\Omega^p_X$ is therefore the cohomology of the complex of global sections, which is the Dolbeault complex. Example: on a compact Riemann surface $X$ of genus $g$, $H^1(X, \mathcal{O}_X) \cong H^{0,1}_{\bar\partial}(X)$, and both have dimension $g$. For $p = 0$, the theorem identifies the cohomology of the structure sheaf $\mathcal{O}_X = \Omega^0_X$ with the obstruction groups to solving $\bar\partial u = f$.

Going deeper

For a student

The isomorphism is the complex analogue of de Rham's theorem, which computes the cohomology of the constant sheaf with smooth forms and the operator $d$. It lets one prove vanishing theorems for sheaf cohomology with analytic tools, such as Hörmander's $L^2$ estimates or harmonic forms. On compact Kähler manifolds, Hodge theory represents each class by a harmonic form, and the isomorphism then gives the Hodge decomposition $H^k(X, \C) \cong \bigoplus_{p+q=k} H^q(X, \Omega^p_X)$.