Hirzebruch–Riemann–Roch

Hirzebruch–Riemann–Roch
Name	Hirzebruch–Riemann–Roch theorem
Field	Algebraic geometry; Differential topology
Introduced	1954
Inventor	Friedrich Hirzebruch

Hirzebruch–Riemann–Roch is a foundational theorem relating topological invariants of complex manifolds to holomorphic and algebraic data, providing an equality between an analytic index and a topological characteristic class pairing. It connects work of Bernhard Riemann, Henri Poincaré, David Hilbert, and André Weil with twentieth-century developments by Friedrich Hirzebruch, Jean-Pierre Serre, and Alexander Grothendieck. The theorem served as a bridge between techniques from Algebraic geometry, Differential topology, K-theory, and representation theory exemplified by connections to Atiyah–Singer index theorem and Grothendieck–Riemann–Roch.

Statement

The theorem asserts that for a compact complex manifold X equipped with a holomorphic vector bundle E, the holomorphic Euler characteristic χ(X,E) equals the evaluation on the fundamental class of X of the cup product of characteristic classes: the Chern character ch(E) and the Todd class td(TX). The statement refines earlier work of Riemann, Roch, and Noether by expressing an index as an integral over X involving Chern classes c_i(TX) and c_j(E) and sits alongside index formulas by Atiyah, Singer, and Grothendieck that use K-theory and cohomological pushforwards.

Historical background and motivation

The theorem grew out of attempts to generalize the classical Riemann–Roch theorem for compact Riemann surfaces studied by Bernhard Riemann and Gustav Roch to higher dimensions, a program influenced by researchers such as Federigo Enriques, Oscar Zariski, and André Weil. Hirzebruch announced the formula in the 1950s after work on characteristic classes initiated by Hirzebruch and building on foundations by Élie Cartan, Hassler Whitney, and Shiing-Shen Chern. Subsequent developments tied the theorem to Serre duality and to spectral methods used by John Nash and later consolidated by Michael Atiyah and Isadore Singer in their index theorem, while Grothendieck framed a far-reaching generalization that unified ideas from Alexander Grothendieck's work on Schemes and Coherent sheaves.

Proofs and methods

Hirzebruch's original proof used topological methods drawing on the theory of characteristic classes developed by Hirzebruch and Chern, together with cobordism ideas influenced by René Thom. Later proofs employed sheaf cohomology and spectral sequences inspired by Jean-Pierre Serre and techniques from Homological algebra introduced by Samuel Eilenberg and Saunders Mac Lane. The analytic approach using elliptic operators and heat kernel methods was developed by Atiyah and Singer, while Grothendieck's algebraic proof leveraged K-theory and the formalism of derived functors advanced by Grothendieck and Jean-Louis Verdier. Further expositions connected the theorem to fixed-point formulas due to Lefschetz and to localization formulas used by Berline and Vergne.

Consequences and applications

The theorem underpins computation of dimensions of spaces of global sections of line bundles appearing in the work of Alexander Grothendieck, David Mumford, and Oscar Zariski on the classification of algebraic varieties; it influences moduli problems studied by Pierre Deligne and William Fulton. In complex differential geometry it informs curvature-based results traced to Shiing-Shen Chern and Shing-Tung Yau, and in mathematical physics the index viewpoint is central to developments by Edward Witten and Michael Atiyah in quantum field theory and string theory. Applications include enumerative results used by Max Noether and computational tools exploited in algorithms influenced by Donald Knuth and John Conway in experimental mathematics.

Examples and computations

For compact Riemann surfaces of genus g the theorem reduces to the classical Riemann–Roch formula proved by Bernhard Riemann and refined by Gustav Roch; explicit computations of χ for line bundles recover the degree–genus relation used by Federigo Enriques and Francesco Severi. For complex projective spaces ℂP^n the Todd class calculations use Chern roots as in computations by Hirzebruch and give dimensions matching those obtained by representation-theoretic counts related to Hermann Weyl and Élie Cartan. For complex tori and abelian varieties the theorem yields formulae applied by André Weil and Igor Shafarevich in the study of Jacobian varieties and in the proof strategies of classical results by Alexander Grothendieck and Pierre Deligne.

Generalizations and related theorems

Grothendieck extended the result to a far-reaching Grothendieck–Riemann–Roch theorem for proper morphisms of Schemes, synthesizing ideas of Grothendieck, Jean-Pierre Serre, and Samuel Eilenberg within K-theory developed by Grothendieck and Michael Atiyah. The analytic index perspective culminated in the Atiyah–Singer index theorem connecting elliptic operators to topological indices, and equivariant versions were developed by Berline, Atiyah, and Vergne leading into localization techniques widely used by Edward Witten. Further extensions include versions for complex analytic spaces treated by Harrison and for noncommutative geometry inspired by Alain Connes and Maxim Kontsevich.

Category:Algebraic geometry Category:Differential topology Category:Theorems in mathematics