Riemann–Roch theorem
| Field | Algebraic geometry and complex analysis |
|---|---|
| First proof by | Gustav Roch |
| First proof in | 1865 |
| Generalizations | Atiyah–Singer index theorem Grothendieck–Riemann–Roch theorem Hirzebruch–Riemann–Roch theorem Riemann–Roch theorem for surfaces Riemann–Roch-type theorem |
| Consequences | Clifford's theorem on special divisors Riemann–Hurwitz formula |
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings.
Initially proved as Riemann's inequality by Riemann (1857), the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student Gustav Roch (1865). It was later generalized to algebraic curves, to higher-dimensional varieties and beyond.