Suppose $X$ is a Riemann surface, $Y$ is a Hausdorff topological space and $p: Y\to X$ is a local homeomorphism. Then there is a unique complex structure on $Y$ such that $p$ is holomorphic. Now if $\pi:Y\to X$ is a branched cover, can we also lift a complex structure to $Y$, s.t. $\pi$ is holomorphic?
closed as unclear what you're asking by Moishe Kohan, Yanior Weg, Xander Henderson, Adrian Keister, José Carlos Santos May 3 at 18:17
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