= Laurent gluing of Dolbeault primitives on the projective line
{c}
{title2=$H^{0,1}(\mathbb P^1)=0$}
For a smooth $(0,1)$-form on the <complex projective line>, solve its <Dolbeault operator> equation on a disc about zero and a disc about infinity. The difference of the primitives is a <holomorphic function> on their annular overlap. Split its <Laurent series> into a nonnegative-power part extending to the finite disc and a negative-power part extending holomorphically across infinity. Subtract the first part from the finite primitive and add the second to the infinite primitive. The results agree and define a global primitive, without <Hodge theory>.
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