Riemann–Roch

Riemann–Roch
Name	Riemann–Roch
Field	Algebraic geometry
Introduced	19th century
Authors	Bernhard Riemann; Gustav Roch
Keywords	Algebraic curve, divisor, line bundle, sheaf cohomology, Euler characteristic

Riemann–Roch is a central family of theorems in algebraic geometry relating analytical, topological, and algebraic invariants of algebraic varieties via linear systems, sheaf cohomology, and characteristic classes. Originating in 19th-century work by Bernhard Riemann and extended by Gustav Roch, its generalizations were developed by Hirzebruch, Grothendieck, and others, influencing research in complex analysis, number theory, mathematical physics, and representation theory.

History and motivation

The origin lies in contributions by Bernhard Riemann and Gustav Roch concerning compact Riemann surfaces and meromorphic functions, motivated by problems studied in correspondence with Carl Friedrich Gauss and in the milieu of Felix Klein and Henri Poincaré. Subsequent 20th-century advances involved figures such as André Weil, whose work connected to the Weil conjectures, and Oscar Zariski, who approached algebraic surfaces, while Federico Enriques and Francesco Severi contributed to classification problems that raised questions resolved by higher-dimensional analogues. The formulation in terms of sheaf cohomology owes much to Jean-Pierre Serre and the development of homological methods by Alexander Grothendieck and collaborators at the Institut des Hautes Études Scientifiques and the École Normale Supérieure.

Classical Riemann–Roch theorem (curves)

For a compact Riemann surface or smooth projective curve over a field, the classical statement relates the dimension of spaces of sections of a line bundle to degree and genus. The curve case was first articulated by Bernhard Riemann and proved in a refined form by Gustav Roch; it is central to the work of Karl Weierstrass on abelian functions and to the theory of Jacobian varietys advanced by Niels Henrik Abel and Carl Jacob Jacobi. The theorem underpins the construction of the Abel–Jacobi map and the study of special divisors in the work of George Kempf and David Mumford, and it figures in classification results connected to Noether's theorem in algebraic geometry and to moduli problems studied by Alexander Grothendieck and Pierre Deligne.

Riemann–Roch for surfaces and higher-dimensional varieties

Extensions to surfaces were pursued by Federico Enriques and Francesco Severi and were systematized with topological input by Friedrich Hirzebruch, who applied characteristic classes such as the Todd class and Chern class to relate holomorphic Euler characteristics to intersection theory. The surface case influenced the birational classification program of Kunihiko Kodaira and later the minimal model program associated with Shigefumi Mori and Vyacheslav Shokurov. Higher-dimensional formulations demanded the machinery developed by Alexander Grothendieck and Jean-Louis Verdier and connected to work by Michael Artin and Paul Vojta on arithmetic analogues and to considerations in string theory explored by Edward Witten.

Hirzebruch–Riemann–Roch and Grothendieck–Riemann–Roch

The Hirzebruch–Riemann–Roch theorem, proved by Friedrich Hirzebruch, provided a bridge between complex analytic index theorems such as the Atiyah–Singer index theorem of Michael Atiyah and Isadore Singer and algebraic methods; it introduced characteristic-class calculations that connect to the Todd genus. Grothendieck formulated a far-reaching generalized Riemann–Roch in the language of K-theory and derived functors, a statement proved in the context of the Séminaire de Géométrie Algébrique under Grothendieck's leadership and elaborated by Jean-Pierre Serre, Pierre Deligne, and Luc Illusie. These results link to the development of algebraic K-theory by Daniel Quillen and to applications in motivic cohomology and the study of Lefschetz fixed-point theorem analogues in arithmetic geometry.

Applications and examples

Applications include the computation of dimensions of linear systems on curves used in the work of André Weil on algebraic curves over finite fields and in the proof of the Riemann hypothesis for curves via the Weil conjectures proved by Pierre Deligne. Concrete examples arise in the study of plane curves considered by Bernhard Riemann and Karl Weierstrass, in analyses of canonical models exploited by Kunihiko Kodaira and David Mumford, and in enumerative problems linked to Hermann Schubert and to modern approaches by Maxim Kontsevich. In mathematical physics, Riemann–Roch-type formulas appear in conformal field theory studied by Alexander Zamolodchikov and in string-theoretic index calculations by Edward Witten.

Proofs and methods

Proofs in the classical case use period matrices and Abelian integrals developed by Niels Henrik Abel and Carl Friedrich Gauss and analytic techniques refined by Riemann and Gustav Roch. Sheaf-theoretic proofs follow the approach of Jean-Pierre Serre using coherent cohomology and duality theorems related to work by Jean-Louis Verdier and Alexander Grothendieck. Topological and index-theoretic proofs leverage the Atiyah–Singer index theorem and techniques introduced by Friedrich Hirzebruch; categorical and K-theoretic proofs originate in Grothendieck's seminar and were furthered by Daniel Quillen and Jean-Louis Verdier for derived categories and for treatment of singular varieties in the work of Luc Illusie and Joseph Bernstein.

Category:Algebraic geometry