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.)



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


