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When does the complement of a hypersurface in a projective space admit a nonconstant holomorphic function?

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This should have probably been reflected towards math.stachexchange.com or another similar site... – Mariano Suárez-Alvarez May 10 '11 at 1:50

Always. If $H=Z(f)$ with $\deg f=d$, then $\dfrac gf$ is a non-constant holomorphic function on $\mathbb P^n\setminus H$ for any homogenous $g\neq \lambda\cdot f$ of $\deg g=d$.

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