Let $X$, $Y$ be complex affine algebraic manifolds (closed submanifolds of $\mathbb{C}^n$), let $f\colon Y\to X$ be a finite covering. Let $\mathcal{L}$ be a holomorphic line bundle on $X$. Suppose $f^*\mathcal{L}$ is an algebraic line bundle. Is $\mathcal{L}$ necessarily algebraic?

(This is true when $f$ admits a section. In general $f_*\mathcal{O}_X$ is locally free, we can recover $\mathcal{L}$ Zariski locally (by the projection formula $f_*\mathcal{L}\cong f_*\mathcal{O}_Y\otimes\mathcal{L}$, then read the factors over $U_i$ over which $f_*\mathcal{O}_{Y}$ is locally free, using $f_*\mathcal{O}_Y|_{f^{-1}(U_i)}\cong\mathcal{O}_{U_i}^{\oplus d}$), but I am not sure if the transitions can be chosen algebraically.)


1 Answer 1


Yes. This follows from P. Deligne "Equations differentielles..." LNM 163, Proposition II 2.22, since the notion of moderate growth at infinity will coincide for $X$ and $Y$. This works more generally for any coherent sheaf.


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