 Dolbeault cohomology of the projective line

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.