Hirzebruch–Riemann–Roch theorem

Hirzebruch–Riemann–Roch theorem
Name	Hirzebruch–Riemann–Roch theorem
Field	Geometry, Topology, Algebraic geometry
Introduced	1954
Author	Friedrich Hirzebruch
Related	Riemann–Roch theorem, Atiyah–Singer index theorem, Grothendieck–Riemann–Roch theorem

Hirzebruch–Riemann–Roch theorem The Hirzebruch–Riemann–Roch theorem provides a formula relating holomorphic Euler characteristics of holomorphic vector bundles on compact complex manifolds to topological characteristic classes, forming a bridge between Friedrich Hirzebruch's work in Germany and later developments by Michael Atiyah and Isadore Singer. It refines the classical Riemann–Roch theorem for Riemann surfaces and anticipates the Grothendieck–Riemann–Roch theorem through connections with Alexander Grothendieck, Jean-Pierre Serre, and the foundations of sheaf theory. The theorem played a central role in the interaction among Algebraic geometry, Differential topology, and K-theory.

Statement of the theorem

For a compact complex manifold X and a holomorphic vector bundle E on X, Hirzebruch gave an equality expressing the holomorphic Euler characteristic χ(X,E) in terms of the Chern character and the Todd class. In modern form, the theorem equates χ(X,E) = ⟨ch(E)·td(TX), [X]⟩, linking the analytic invariant χ(X,E) with topological invariants coming from Chern classes and the tangent bundle TX, and thus relating ideas from Friedrich Hirzebruch, Bernhard Riemann, Gustav Roch, Jean Leray, Henri Cartan and Oscar Zariski.

Historical background and motivation

Hirzebruch announced his theorem in 1954 during a period shaped by work of Bernhard Riemann and Gustav Roch on algebraic curves and the mid-20th-century expansion of Algebraic geometry led by figures such as Oscar Zariski and André Weil. The motivation came from computing dimensions of spaces of holomorphic sections, a theme present in the contributions of Ernst Kähler, Kunihiko Kodaira, and Jean-Pierre Serre; Hirzebruch synthesized these with ideas from Cobordism theory and topologists like René Thom and John Milnor. Subsequent reinterpretations by Michael Atiyah and Isadore Singer linked the theorem to K-theory developed by Alexander Grothendieck and John von Neumann's operator theory, while Grothendieck later generalized it within his work on Derived categories and the Grothendieck–Riemann–Roch theorem.

Chern character, Todd class, and index formula

The formula uses the Chern character ch(E) and the Todd class td(TX) of the complex manifold's tangent bundle TX; both are constructed from Chern classes which trace back to work of Shiing-Shen Chern and Élie Cartan. The pairing ⟨·,[X]⟩ with the fundamental homology class [X] yields an integer equal to the holomorphic Euler characteristic χ(X,E) computed analytically via Dolbeault cohomology, a tool advanced by Georges de Rham, Kiyoshi Oka, and Kunihiko Kodaira. This index-type identity presaged the Atiyah–Singer index theorem where Michael Atiyah and Isadore Singer provided an analytical framework linking elliptic differential operators studied by Lars Hörmander and Atle Selberg to topological invariants.

Proofs and approaches

Hirzebruch's original proof combined topological methods prominent in the work of René Thom and calculations in cobordism employing tools later systematized by Milnor and Stasheff. Alternate proofs use analytic techniques from Kodaira's vanishing theorems and Hodge theory developed by W. V. D. Hodge and Atiyah, while Grothendieck's approach recasts the statement using K-theory and Grothendieck groups of coherent sheaves, drawing on foundations by Jean Leray and Alexander Grothendieck. The Atiyah–Singer framework interprets Hirzebruch's formula as a special case of an index theorem for elliptic operators, connecting to operator algebra perspectives championed by Israel Gelfand and John von Neumann.

Examples and applications

On a compact Riemann surface of genus g, the theorem recovers the classical degree–genus formula attributed to Bernhard Riemann and Gustav Roch and informs the work of Felix Klein and Hermann Weyl on moduli spaces; in projective algebraic geometry, it computes dimensions of spaces of global sections on projective varieties as in experiments by David Mumford, Pierre Deligne, and Jean-Pierre Serre. Applications extend to enumerative problems tackled by Alexander Grothendieck and Igor Dolgachev, to characteristic class computations used by René Thom and Raoul Bott, and to string-theoretic index counts in perspectives influenced by Edward Witten and Michael Green.

Generalizations and related results

Generalizations include the Grothendieck–Riemann–Roch theorem formulated by Alexander Grothendieck, which extends Hirzebruch's formula to proper morphisms between schemes and leverages the Grothendieck group of coherent sheaves, and the Atiyah–Singer index theorem by Michael Atiyah and Isadore Singer, which frames the result in the context of elliptic operators. Further developments connect to the work of Maxim Kontsevich on deformation quantization, to categorical upgrades by Jacob Lurie and Dennis Gaitsgory, and to arithmetic analogues explored by Jean-Pierre Serre and Alexander Beilinson.

Category:Theorems in differential geometry