= Solution
We use complex-valued smooth <differential forms>. The <sheaf of smooth differential forms> is $\mathcal A^k(U)=\Gamma(U,\bigwedge^k(T^*X\otimes\mathbb C))$, with the usual restriction maps. Dualizing the <type decomposition of the complexified tangent bundle> and taking <exterior powers> decomposes this bundle into the summands
$$
\bigwedge^p(T^{1,0}X)^*\otimes\bigwedge^q(T^{0,1}X)^*,\qquad p+q=k.
$$
Their smooth sections form the <sheaf of differential forms of type (p, q)> $\mathcal A^{p,q}$. Locally a section is a sum of $f_{I,J}\,dz_I\wedge d\bar z_J$ with $|I|=p$, $|J|=q$ and smooth coefficients. Holomorphic transition maps preserve types, so the local decompositions agree globally. Hence
$$
\boxed{\mathcal A^k(X)=\bigoplus_{p+q=k}\mathcal A^{p,q}(X).}
$$
The <exterior derivative> splits as $d=\partial+\bar\partial$, with bidegrees $(1,0)$ and $(0,1)$. Its square being zero gives $\partial^2=\bar\partial^2=0$ and $\partial\bar\partial+\bar\partial\partial=0$. The <Dolbeault cohomology> is therefore
$$
\boxed{H^{p,q}_{\bar\partial}(X)=
\frac{\ker(\bar\partial:\mathcal A^{p,q}(X)\to\mathcal A^{p,q+1}(X))}
{\bar\partial\mathcal A^{p,q-1}(X)}.}
$$
Forms in negative or out-of-range bidegrees are understood to be zero.
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