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Any non-constant surjective holomorphic map between connected compact complex manifolds of equal dimension is a ramified finite-sheet covering. If this map is in particular bijective, then there is only one sheet, and thus is a biholomorphism.
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Any non-constant map between compact complex manifolds of equal dimension is a ramified finite-sheet covering. If this map is in particular bijective, then there is only one sheet, and thus is a biholomorphism.