= Solution
Let $\mathcal A^{p,q}(X)$ be the space of smooth complex <differential forms> of type $(p,q)$. The <Dolbeault operator> satisfies $\bar\partial^2=0$, and the <Dolbeault cohomology> is
$$
\boxed{H_{\bar\partial}^{p,q}(X)=
\frac{\ker(\bar\partial:\mathcal A^{p,q}\to\mathcal A^{p,q+1})}
{\operatorname{im}(\bar\partial:\mathcal A^{p,q-1}\to\mathcal A^{p,q})}.}
$$
For $q=0$ the denominator is zero; forms outside the dimension range are zero.
A sequence of <sheaves> is an <exact sequence of sheaves> precisely when its sequence of <stalks> at every point is exact. Here this means the first map is injective on <stalks>, the last is surjective on <stalks>, and the image equals the kernel at the middle <stalk>. In particular, a section of the last <sheaf> need only have local lifts; surjectivity on all global sections is not required.
The associated <long exact sequence in sheaf cohomology> of <Čech cohomology>, understood in the direct limit over open covers, is
$$
\begin{aligned}
0&\longrightarrow\check H^0(X,\mathcal F)
\longrightarrow\check H^0(X,\mathcal E)
\longrightarrow\check H^0(X,\mathcal G)
\xrightarrow{\delta_0}\check H^1(X,\mathcal F)\longrightarrow\cdots\\
&\longrightarrow\check H^q(X,\mathcal F)
\longrightarrow\check H^q(X,\mathcal E)
\longrightarrow\check H^q(X,\mathcal G)
\xrightarrow{\delta_q}\check H^{q+1}(X,\mathcal F)\longrightarrow\cdots.
\end{aligned}
$$
The degree-zero groups are global sections. The connecting map is obtained by locally lifting a <Čech cocycle> to the middle <sheaf> and taking its <Čech coboundary>, which takes values in the first <sheaf>. Different lifts change it by a <Čech coboundary>.
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