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Let $X$ be a projective variety and let $D$ be an ample divisor. Is the complement of the support of $D$ in $X$ affine?

We can suppose $D$ is very ample. (Just replace it with a multiple.) I'm trying to reduce to the case of a hyperplane on projective space, but I can't do it.

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up vote 7 down vote accepted

Yes. As you say we may assume $D$ is very ample. So there is a closed immersion of $X$ into $\mathbb{P}^n$ such that $D$ is $X \cap H$ for some hyperplane $H$. Pick coordinates $[x_0: x_1: \ldots: x_n]$ for the ambient $\mathbb{P}^n$ such that the vanishing locus of $x_0$ is $H$. Let $f_1, \ldots, f_m$ be homogeneous equations cutting out $X$ in these coordinates. Then their dehomogenizations with respect to $x_0$ are homogeneous equations cutting out $X \setminus D$ in $\mathbb{P}^n \setminus H = \mathbb{A}^n$.

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