Let $X$ be a complex manifold (e.g., a complex torus), $Z$ an analytic subset of $X$ with codimension $\ge 2$. Let $U$ be the complement of $Z$ in $X$. I wonder if any holomorphic line bundle on $U$ can be extended to a holomorphic line bundle on $X$. I would greatly appreciate a reference (if such an extension always exists. The case of codimension $\ge 3$ follows from results of R. Harvey, Amer. J. Math. 96 (1974), 498--504.).
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$\begingroup$ Note that if such an extension exists, then it is necessarily unique in the sense that any two extensions are canonically (and uniquely) isomorphic. $\endgroup$– André HenriquesApr 21, 2019 at 23:27
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3$\begingroup$ I don't think it's true with the hypothesis you want. See mathoverflow.net/questions/35788/… $\endgroup$– Donu ArapuraApr 21, 2019 at 23:59
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3$\begingroup$ If the line bundle is semipositive, then yes. This is a result of MR0289806 Shiffman, Bernard Extension of positive holomorphic line bundles. Bull. Amer. Math. Soc. 77 1971 1091–1093. Without semipositivity or codimension at least 3, it may be impossible, as in the question linked to by Donu Arapura. $\endgroup$– Margaret FriedlandApr 22, 2019 at 0:44
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