Solution
= Solution
Cover the product by $U_0=\mathbb C_z\times\mathbb C_w$ and $U_\infty=\mathbb C_\zeta\times\mathbb C_w$, with $\zeta=1/z$ on their intersection $\mathbb C^*\times\mathbb C$. The permitted vanishing and the <Dolbeault theorem> make this an <acyclic cover> for both $\mathcal O$ and $\Omega^2$. The <acyclic cover theorem> lets its two-term Čech complex compute the cohomology. In particular, all groups with $q\geq2$ vanish.
For $p=0$, a global <holomorphic function> is constant on each compact <complex projective line> fibre, by the <maximum modulus principle>. Its remaining dependence on $w$ is entire. Hence $H^{0,0}_{\bar\partial}=\mathcal O(\mathbb C)$. For the first Čech group, a function on the intersection has a <Laurent series>
$$
F(z,w)=\sum_{j\in\mathbb Z}a_j(w)z^j.
$$
Each coefficient is entire in $w$ by its <Cauchy integral formula>. The nonnegative powers extend to $U_0$, and the negative powers extend to $U_\infty$ in coordinate $\zeta$. These two series converge locally uniformly with the parameter $w$, by the usual Laurent estimates on compact parameter sets. Thus every intersection function is a <Čech coboundary> and $H^{0,1}_{\bar\partial}=0$.
For $p=2$, write a two-form on the intersection as $F(z,w)dz\wedge dw$. The other chart has
$$
d\zeta\wedge dw=-z^{-2}dz\wedge dw.
$$
Forms extending from $U_0$ have coefficient powers $j\geq0$; those extending from $U_\infty$ have powers $j\leq-2$. A globally defined two-form would need to have both types of expansion, so it is zero. In the first Čech quotient, precisely the $z^{-1}$ term remains, and its coefficient is an arbitrary <entire function> of $w$. Consequently
$$
\boxed{
H_{\bar\partial}^{p,q}(\mathbb P^1\times\mathbb C)\cong
\begin{cases}
\mathcal O(\mathbb C),&(p,q)=(0,0),\\
\mathcal O(\mathbb C),&(p,q)=(2,1),\\
0,&p\in\{0,2\}\text{ and all other }q.
\end{cases}}
$$
The second nonzero group is represented in <Čech cohomology> by $a(w)z^{-1}dz\wedge dw$. This explicit residue description is the <Dolbeault cohomology of the projective line times the affine line>; it also identifies that group naturally with the holomorphic one-forms on the affine factor.