# "Holomorphic line bundle" + "algebraic after finite cover" implies "algebraic"?

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.)

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.