Riemann–Hurwitz formula

The Riemann–Hurwitz formula relates the genera of compact Riemann surfaces under a nonconstant holomorphic map, correcting for its ramification.

Holomorphic map: A complex-differentiable map between complex manifolds. The formula applies to nonconstant holomorphic maps between compact Riemann surfaces.
Compact Riemann surface: A compact connected one-dimensional complex manifold. The formula compares the genera of the source and target surfaces.
Branched covering: A map that is a covering away from a specified set of branch values. The formula quantifies how branching alters the topology of a covering surface.
Bernhard Riemann: A German mathematician whose work founded major parts of modern analysis and geometry. Riemann’s theory of surfaces supplied the setting for the genus relation.
Degree of a holomorphic map: The number of preimages of a generic point under a nonconstant holomorphic map of compact Riemann surfaces. The degree multiplies the target’s Euler characteristic in the formula.
Genus: A nonnegative integer measuring the number of handles on a compact connected orientable surface. The formula expresses the source genus in terms of the target genus and map data.
Hyperelliptic curve: A smooth algebraic curve admitting a degree-two map to the projective line. Counting its branch points with the formula determines its genus.
Adolf Hurwitz: A German mathematician known for work in algebra, number theory, and the theory of Riemann surfaces. Hurwitz developed the formula’s systematic form for branched maps.
Ramification point: A point where a holomorphic map is locally equivalent to the power map z ↦ zᵉ for e greater than one. These points are precisely where the map fails to be locally one-to-one.
Holomorphic differential: A holomorphic section of the cotangent bundle of a complex manifold. The dimension of holomorphic differentials on a compact Riemann surface equals its genus.

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Algebraic functionRelated: It quantifies how branch points shape the surface of an algebraic function.