= Dolbeault cohomology of the projective line times the affine line
{c}
{title2=$H^{0,0}=\mathcal O(\mathbb C),\quad H^{2,1}=\mathcal O(\mathbb C)$}
For $X=\mathbb P^1\times\mathbb C$, the two standard affine charts and their intersection form an <acyclic cover> for the <sheaves> of holomorphic forms. Their <Čech cohomology> gives $H^{0,0}_{\bar\partial}=\mathcal O(\mathbb C)$ and $H^{2,1}_{\bar\partial}=\mathcal O(\mathbb C)$, with all other groups for $p\in\{0,2\}$ zero. For the top-form <sheaf>, the transition $d(1/z)=-z^{-2}dz$ removes every <Laurent series> power except $z^{-1}$ from the first cohomology quotient; its coefficient may be an arbitrary <entire function> on the affine factor.
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