= Dolbeault cohomology of the projective line
{c}
{title2=$H^{0,1}(\mathbb{CP}^1,\mathcal O)=0$}
The <Dolbeault theorem> identifies this group with first <Čech cohomology> of the trivial holomorphic line bundle. On the two standard affine charts, a holomorphic overlap function has a <Laurent series>. Its nonnegative powers extend to the chart at zero, and its negative powers extend to the chart at infinity. Every overlap cocycle is therefore a coboundary. Consequently the <Dolbeault operator> from functions to $(0,1)$-forms is <surjective>.
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