On the GAGA principle for algebraic affine hypersurfaces

Abstract

For any complete $\mathbb{C}$-algebraic variety Y and its underlying compact $\mathbb{C}$-analytic space $\mathcal{Y}$, it follows from the well known GAGA principle that the algebraic Picard group $Pic(Y)$ and the analytic Picard group $\mathbb{P}\!ic(\mathcal{Y})$ are isomorphic. Our main purpose here is to provide a simple proof of an analogous situation for non complete $\mathbb{C}$-algebraic varieties, namely $\mathbb{C}$-algebraic affine hypersurfaces with at most isolated singularities.