---
tag: 5858M5PGN6
kind: statement
role: lemma
name: Dolbeault–Grothendieck lemma
msc: [32W05, 32A10, 58A10]
uses: [DJBWP7D3M8, DC81TCR98Y, M1VKCW3RFW]
about: [M1VKCW3RFW]
sources:
  - url: https://en.wikipedia.org/wiki/Dolbeault_cohomology
---

## Statement

Let $\Delta = \set{z \in \C^n : \abs{z_j} < r_j \text{ for } j = 1, \dots, n}$ be a [[polydisc|DJBWP7D3M8]] with radii $0 < r_j \le \infty$.
Let $p \ge 0$ and $q \ge 1$ be integers, and let $\alpha$ be a smooth [[(p,q)-form|DC81TCR98Y]] on $\Delta$ with $\bar\partial \alpha = 0$, where $\bar\partial$ is the [[dbar operator|M1VKCW3RFW]].
Then there is a smooth $(p,q-1)$-form $\beta$ on $\Delta$ with $\bar\partial \beta = \alpha$.

## Remarks

The key case is $n = 1$, $q = 1$: for a smooth function $g$ with compact support in $\C$, the function
$$u(z) = \frac{1}{2\pi i} \int_\C \frac{g(w)}{w - z}\, dw \wedge d\bar w$$
is smooth and satisfies $\partial u / \partial \bar z = g$.
In general one first proves the result on a slightly smaller polydisc, by induction on the largest index $k$ such that some $d\bar z_k$ occurs in $\alpha$.
The coefficients involving $d\bar z_k$ are holomorphic in $z_{k+1}, \dots, z_n$, and solving the one-variable equation in $z_k$ with a cutoff removes $d\bar z_k$ without destroying this holomorphy.
An exhaustion of $\Delta$ by smaller polydiscs, together with Runge approximation to correct the successive solutions, gives the result on $\Delta$ itself.
Example: on $\C$, the $(0,1)$-form $\bar z \, d\bar z$ is $\bar\partial$ of $\bar z^2/2$.
The condition $q \ge 1$ is necessary to make the statement meaningful; for $q = 0$ a $\bar\partial$-closed $(p,0)$-form is a holomorphic $p$-form and need not vanish.
The conclusion fails on general open sets: on $\C^2 \setminus \set{0}$ the Dolbeault group $H^{0,1}$ is not zero.

## Explanations

### L1 human

The lemma says that on a polydisc every form with $\bar\partial \alpha = 0$ is of the form $\bar\partial \beta$.
It is the complex analogue of the Poincaré lemma, which says that closed forms on a ball are exact.
In particular, on a small polydisc around any point of a complex manifold, the equation $\bar\partial \beta = \alpha$ can always be solved.

### L2 human

The lemma is the local input to the Dolbeault isomorphism.
It says that the sequence of sheaves $0 \to \Omega^p \to \mathcal{A}^{p,0} \to \mathcal{A}^{p,1} \to \cdots$ is exact, where $\Omega^p$ is the sheaf of holomorphic $p$-forms and $\mathcal{A}^{p,q}$ is the sheaf of smooth $(p,q)$-forms.
So every Dolbeault cohomology class is locally trivial, and the global groups measure only the topology and the complex structure of the manifold as a whole.