The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex geometry that relates the properties of a branched cover of Riemann surfaces (or algebraic curves) to the properties of its base surface and the branching behavior of the cover.
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Let be a nonconstant holomorphic map between compact Riemann surfaces that are connected spaces, without boundary. Write for their genera and for the degree of a holomorphic map. With analytic ramification index at each , the formula isHere is the Euler characteristic. Only finitely many ramification points contribute. The holomorphic map is automatically surjective; the valency theorem gives even at a branch value of a holomorphic map. Constant maps are excluded.[ref]
The useful interpretation is that total ramification of a holomorphic map measures how much the source Euler characteristic falls short of copies of the target. The ramification of a power map of the Riemann sphere spends that entire correction at two points; the double cover of the sphere with four branch values spends it at four points and produces a genus-one surface.