Grothendieck–Riemann–Roch

Grothendieck–Riemann–Roch
Name	Grothendieck–Riemann–Roch
Field	Algebraic geometry
Introduced	1950s–1960s
Author	Alexander Grothendieck, Jean-Pierre Serre, Michael Atiyah, Friedrich Hirzebruch
Related	Riemann–Roch theorem, Hirzebruch–Riemann–Roch, K-theory, Chow ring

Contents

Grothendieck–Riemann–Roch Grothendieck–Riemann–Roch is a foundational theorem in algebraic geometry connecting Alexander Grothendieck, Jean-Pierre Serre, Michael Atiyah, Friedrich Hirzebruch and others' work on K-theory, Chow group, Chern character and Todd class; it generalizes the Riemann–Roch theorem, the Hirzebruch–Riemann–Roch theorem and the Riemann–Hurwitz formula. The theorem provides a natural transformation between Grothendieck group of coherent sheaves and the Chow ring cohomology theory for morphisms of schemes and has influenced research by Jean-Louis Verdier, Pierre Deligne, Alexander Beilinson, and William Fulton.

Introduction

The theorem arose from efforts by Friedrich Hirzebruch to unify results of Bernhard Riemann and Gustav Roch and was reformulated by Jean-Pierre Serre and expanded by Alexander Grothendieck in the context of schemes. Grothendieck's formulation employed tools from Algebraic K-theory, developed later by Daniel Quillen and influenced by John Milnor, and used intersection theoretic ideas later axiomatized by William Fulton and Pierre Samuel. Historical developments connect to work of Oscar Zariski, André Weil, Alexander Grothendieck's school at Institut des Hautes Études Scientifiques, and later expositions by Robin Hartshorne.

Let f: X → Y be a proper morphism of smooth quasi-projective schemes over a field; let K0(X) denote the Grothendieck group of coherent sheaves, and let A*(X) denote the Chow ring of X. The theorem asserts ch(f_!(α)) · Td(Y) = f_*(ch(α) · Td(X)) for α in K0(X), where ch is the Chern character and Td is the Todd class; here f_! denotes the pushforward in K-theory and f_* denotes the pushforward in Chow group (or cohomology ring). In analytic contexts one replaces scheme by complex manifold and uses topological K-theory; comparable formulations were given by Michael Atiyah and Isadore Singer in relation to the Atiyah–Singer index theorem. The statement subsumes classical results such as Hirzebruch–Riemann–Roch, Riemann–Hurwitz formula, Noether formula, and the Lefschetz fixed-point theorem in various settings.

Grothendieck's original proof used the machinery of derived category, derived functors, and the formalism of K-theory developed in seminars at Séminaire de Géométrie Algébrique under Grothendieck and collaborators including Jean-Louis Verdier and Pierre Deligne. Alternative proofs exploit the deformation to the normal cone as in work by William Fulton and Robert MacPherson, or use cobordism theory and complex oriented cohomology by Daniel Quillen and André Joyal. Analytic proofs derive from the Atiyah–Singer index theorem and heat kernel techniques developed by Paul Atiyah, Isadore Singer, Bertram Kostant, and Michael Atiyah's collaborators; Hodge-theoretic approaches involve contributions by Phillip Griffiths and Wilfried Schmid. Later categorical and motivic proofs connect to Beilinson's and Vladimir Voevodsky's work on motives and to Algebraic K-theory of Quillen.

For a smooth projective curve C over a field, applying the theorem to the structure sheaf recovers the classical Riemann–Roch theorem for divisors and relates to the Jacobian variety and the Abel–Jacobi map; these connections involve figures such as Niels Abel and Carl Gustav Jacob Jacobi. For a smooth projective surface S, the theorem yields the Noether formula and links to the classification of algebraic surfaces developed by Kunihiko Kodaira and Enriques. In the case of a finite morphism of curves one obtains the Riemann–Hurwitz formula familiar from work by Heinrich Weber and Bernhard Riemann. The Hirzebruch signature theorem and the Gauss–Bonnet theorem arise as analytic analogues in the presence of a complex structure, aligning with contributions by Hermann Weyl and Henri Poincaré.

Grothendieck–Riemann–Roch underpins computations of intersection numbers central to enumerative problems studied by Oscar Zariski, Federigo Enriques, and modern enumerative geometers like Maxim Kontsevich and Gromov–Witten theory developers. It informs the study of moduli spaces such as Moduli space of curves explored by David Mumford and John Harris, and of vector bundles as in the work of Ramanan, Seshadri, and Alexander Grothendieck. In arithmetic geometry, the theorem has implications for arithmetic intersection theory advanced by Serge Lang, Gerd Faltings, and Shouwu Zhang and relates to the arithmetic Riemann–Roch of Quillen and Gillet–Soulé. It also plays a role in representation-theoretic contexts influenced by Harish-Chandra, George Lusztig, and categorical formulations connected to Bernstein and Beilinson–Bernstein localization.

Generalizations include versions for singular varieties developed by William Fulton and Robert MacPherson using Chern–Schwartz–MacPherson class, equivariant versions by Atiyah–Segal and Michel Brion, and arithmetic refinements by Henri Gillet and Soulé. Noncommutative and derived enhancements connect to Maxim Kontsevich's proposals in homological mirror symmetry and to the Noncommutative geometry program of Alain Connes. Motivic and higher K-theory refinements follow work by Voevodsky, Suslin, Weibel, and Cortiñas, while analytic index-theoretic refinements interplay with the Atiyah–Patodi–Singer index theorem of Isadore Singer and Vladimir Atiyah's collaborators. Contemporary research links to Donaldson–Thomas invariants, Pandharipande–Thomas theory, and categorical developments by Jacob Lurie and Dennis Gaitsgory.