[proofplan]
The proof is local because exactness of a sequence of sheaves is equivalent to exactness on stalks. Around each point of $X$, choose a holomorphic coordinate chart and write smooth $(p,q)$-forms in the standard basis $dz_I \wedge d\bar z_J$. Exactness at $\mathcal A_X^{p,0}$ is the coordinate statement that a smooth $(p,0)$-form has holomorphic coefficients exactly when its $\bar\partial$ vanishes. Exactness in positive antiholomorphic degree follows from the $\bar\partial$-Poincare lemma on coordinate polydiscs, while $\bar\partial^2 = 0$ gives that the displayed sequence is a complex.
[/proofplan]
[step:Reduce sheaf exactness to a local coordinate calculation]
Fix a point $x \in X$. Since exactness of sheaves may be checked on stalks, it is enough to prove that the induced sequence on the stalks at $x$ is exact. Choose a holomorphic coordinate chart $(U,z)$ with $x \in U$, where
\begin{align*}
z: U \to V \subset \mathbb C^n
\end{align*}
is a biholomorphism onto an [open set](/page/Open%20Set) $V$.
For a multi-index $I = (i_1,\dots,i_p)$ with $1 \leq i_1 < \cdots < i_p \leq n$, write
$dz_I := dz_{i_1} \wedge \cdots \wedge dz_{i_p}$. For a multi-index $J = (j_1,\dots,j_q)$ with $1 \leq j_1 < \cdots < j_q \leq n$, write
$d\bar z_J := d\bar z_{j_1} \wedge \cdots \wedge d\bar z_{j_q}$. On $U$, every section
\begin{align*}
\alpha \in \mathcal A_X^{p,q}(U)
\end{align*}
has a unique expression
\begin{align*}
\alpha = \sum_{|I|=p, |J|=q} f_{I,J}\, dz_I \wedge d\bar z_J
\end{align*}
with smooth coefficient functions $f_{I,J}: U \to \mathbb C$.
[/step]
[step:Identify holomorphic forms with smooth $(p,0)$-forms killed by $\bar\partial$]
Let
\begin{align*}
\alpha \in \mathcal A_X^{p,0}(U)
\end{align*}
be written uniquely as
\begin{align*}
\alpha = \sum_{|I|=p} f_I\, dz_I
\end{align*}
with smooth functions $f_I: U \to \mathbb C$. In holomorphic coordinates, the operator
\begin{align*}
\bar\partial: \mathcal A_X^{p,0}(U) \to \mathcal A_X^{p,1}(U)
\end{align*}
acts by
\begin{align*}
\bar\partial \alpha = \sum_{|I|=p} \sum_{j=1}^n \frac{\partial f_I}{\partial \bar z_j}\, d\bar z_j \wedge dz_I.
\end{align*}
The forms $d\bar z_j \wedge dz_I$ form a local smooth frame for the relevant summands of $(p,1)$-forms. Hence $\bar\partial \alpha = 0$ if and only if
\begin{align*}
\frac{\partial f_I}{\partial \bar z_j} = 0
\end{align*}
for every $I$ and every $j \in \{1,\dots,n\}$. By the coordinate characterization of holomorphic functions, this is equivalent to every coefficient function $f_I$ being holomorphic on $U$. Therefore $\ker(\bar\partial: \mathcal A_X^{p,0} \to \mathcal A_X^{p,1})$ is exactly the image of $\Omega_X^p \to \mathcal A_X^{p,0}$.
[guided]
We prove the first nontrivial exactness statement in coordinates. A smooth $(p,0)$-form on $U$ has the form
\begin{align*}
\alpha = \sum_{|I|=p} f_I\, dz_I,
\end{align*}
where each $f_I: U \to \mathbb C$ is smooth and the sum runs over strictly increasing multi-indices $I = (i_1,\dots,i_p)$. The inclusion
\begin{align*}
\Omega_X^p(U) \to \mathcal A_X^{p,0}(U)
\end{align*}
takes a holomorphic $p$-form and regards it as a smooth $(p,0)$-form. Thus its image consists of exactly those smooth $(p,0)$-forms whose coefficient functions are holomorphic.
Now compute $\bar\partial$ on the displayed local expression. Since the coordinate forms $dz_I$ are holomorphic and therefore have no antiholomorphic differential contribution, $\bar\partial$ differentiates only the coefficient functions:
\begin{align*}
\bar\partial \alpha = \sum_{|I|=p} \sum_{j=1}^n \frac{\partial f_I}{\partial \bar z_j}\, d\bar z_j \wedge dz_I.
\end{align*}
The coordinate forms $d\bar z_j \wedge dz_I$ are part of a local smooth frame for $(p,1)$-forms, so the displayed sum vanishes precisely when each coefficient vanishes:
\begin{align*}
\frac{\partial f_I}{\partial \bar z_j} = 0
\end{align*}
for all $I$ and all $j \in \{1,\dots,n\}$. By the coordinate characterization of holomorphic functions, these equations hold exactly when each $f_I$ is holomorphic. Therefore a smooth $(p,0)$-form lies in the kernel of $\bar\partial$ exactly when it is a holomorphic $p$-form. This proves exactness at $\mathcal A_X^{p,0}$.
[/guided]
[/step]
[step:Verify that consecutive $\bar\partial$ maps compose to zero]
Let $q$ be an integer with $0 \leq q \leq n-1$, and let
\begin{align*}
\alpha \in \mathcal A_X^{p,q}(U)
\end{align*}
be written as
\begin{align*}
\alpha = \sum_{|I|=p, |J|=q} f_{I,J}\, dz_I \wedge d\bar z_J.
\end{align*}
Applying $\bar\partial$ twice gives second antiholomorphic derivatives of the smooth coefficient functions $f_{I,J}$. The mixed partial derivatives commute, while the exterior factors $d\bar z_k \wedge d\bar z_j$ are antisymmetric. These two facts cancel term by term, so
\begin{align*}
\bar\partial^2 \alpha = 0.
\end{align*}
Thus the image of each map $\bar\partial: \mathcal A_X^{p,q} \to \mathcal A_X^{p,q+1}$ is contained in the kernel of the next map.
[/step]
[step:Apply the $\bar\partial$-Poincare lemma in positive antiholomorphic degree]
Let $q$ be an integer with $1 \leq q \leq n$, and let the germ
\begin{align*}
[\alpha]_x \in (\mathcal A_X^{p,q})_x
\end{align*}
satisfy $\bar\partial[\alpha]_x = 0$, where for $q=n$ the target sheaf $\mathcal A_X^{p,n+1}$ is zero. Choose a coordinate neighbourhood $W \subset U$ of $x$ whose image $z(W)$ is a polydisc in $\mathbb C^n$, and choose a representative
\begin{align*}
\alpha \in \mathcal A_X^{p,q}(W)
\end{align*}
with $\bar\partial \alpha = 0$ on $W$.
By the $\bar\partial$-Poincare lemma on polydiscs (citing a result not yet in the wiki: Dbar Poincare Lemma), applied to the smooth $\bar\partial$-closed $(p,q)$-form $\alpha$ with $q>0$, after possibly shrinking $W$ to a smaller coordinate neighbourhood $W' \subset W$ of $x$, there exists a smooth $(p,q-1)$-form
\begin{align*}
\beta \in \mathcal A_X^{p,q-1}(W')
\end{align*}
such that
\begin{align*}
\bar\partial \beta = \alpha|_{W'}.
\end{align*}
Therefore
\begin{align*}
[\alpha]_x = \bar\partial[\beta]_x.
\end{align*}
This proves that the kernel of $\bar\partial: (\mathcal A_X^{p,q})_x \to (\mathcal A_X^{p,q+1})_x$ is contained in the image of $\bar\partial: (\mathcal A_X^{p,q-1})_x \to (\mathcal A_X^{p,q})_x$ for every $q>0$.
[/step]
[step:Conclude exactness of the Dolbeault sequence of sheaves]
The first smooth term has kernel equal to $\Omega_X^p$ by the coordinate calculation above. For every positive antiholomorphic degree $q$, the inclusion
\begin{align*}
\operatorname{im}\bigl(\bar\partial: (\mathcal A_X^{p,q-1})_x \to (\mathcal A_X^{p,q})_x\bigr) \subset \ker\bigl(\bar\partial: (\mathcal A_X^{p,q})_x \to (\mathcal A_X^{p,q+1})_x\bigr)
\end{align*}
follows from $\bar\partial^2 = 0$, and the reverse inclusion follows from the local $\bar\partial$-Poincare lemma. Hence the induced sequence on the stalk at every point $x \in X$ is exact. Since exactness of sheaves is equivalent to exactness on all stalks, the displayed sequence of sheaves is exact.
[/step]
