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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 ${\mathbb{C}}^{\ast }\times \mathbb{C}$. 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 $q\ge 2$ 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 $H_{\bar{\partial }}^{0,0}=\mathcal{O}(\mathbb{C})$. For the first Čech group, a function on the intersection has a Laurent seriesEach 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 $H_{\bar{\partial }}^{0,1}=0$.

For $p=2$, write a two-form on the intersection as $F(z,w)dz\wedge dw$. The other chart hasForms extending from $U_0$ have coefficient powers $j\ge 0$; those extending from $U_{\infty }$ have powers $j\le -2$. 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 $w$. ConsequentlyThe 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.