Grothendieck Riemann RochEdit

Grothendieck-Riemann-Roch (GRR) stands as a foundational result in modern algebraic geometry, weaving together K-theory, Chow groups, and characteristic classes. It generalizes the classical Riemann-Roch theorem from curves to higher-dimensional varieties and from individual sheaves to the pushforward along a morphism. In its standard form, GRR relates the Chern character of a pushforward to a Todd-class–weighted pushforward of the original data, encoding both the geometry of the morphism and the geometry of the sheaf.

At its heart, GRR gives a bridge between two languages: the K-theory of coherent sheaves on X and the Chow (or cohomology) theory on Y. For a proper morphism f: X → Y of smooth varieties over a field and a vector bundle E on X, the theorem can be stated as ch(f_! E) = f_( Td(T_f) ∙ ch(E) ), where: - f_! E denotes the alternating sum ∑ (-1)^i [R^i f_ E] in the Grothendieck group K_0(Y), - ch(E) is the Chern character of E in CH^(X) ⊗ Q, - Td(T_f) is the Todd class of the relative tangent bundle T_f, - f_ is the pushforward map on Chow groups (or, after appropriate identifications, on cohomology).

In words: the left-hand side encodes the cohomological content of the direct images of E, while the right-hand side packages the geometry of X relative to Y (via T_f) and the intrinsic geometry of E (via ch(E)) into a single pushforward on Y. The equality holds after tensoring with Q, reflecting the rational nature of the Chern character and the Todd class.

Origins and formulation

GRR sits in the Grothendieckian program to extend elementary index and characteristic-class formulas to the realm of algebraic geometry. It builds on the development of K-theory (the Grothendieck group K_0) and the systematic use of derived functors, as well as the refinement of intersection theory captured in Chow groups. The theorem was proven as part of Grothendieck’s broad program and has since been developed and reformulated in multiple closely related frameworks, notably by Fulton in the setting of schemes and by various authors who study the bivariant theories underlying the six-functor formalism.

Key ingredients include: - K-theory, via the group K_0(X) generated by vector bundles on X and their relations, - The Chern character map ch: K_0(X) ⊗ Q → CH^(X) ⊗ Q, which translates K-theoretic data into Chow-theoretic (or cohomological) data, - The Todd class Td(T_f) of the relative tangent bundle T_f, which encodes the geometric twisting introduced by the morphism f, - The pushforward f_ on Chow groups, which aggregates local data along the fibers of f.

Special cases and intuition

  • Classical Riemann-Roch for curves: If X is a smooth projective curve and f: X → Spec(k) is the structure morphism, GRR reduces to the Hirzebruch-Riemann-Roch formula χ(X, E) = deg(ch(E) ∙ Td(T_X)) integrated over X, recovering the original genus and degree calculations one encounters in the study of line bundles on curves.
  • Curves to points and line bundles: For a line bundle L on X, GRR recovers the familiar relations governing h^0 and h^1 in terms of degrees and genera, generalized to higher-dimensional X via the same formalism.
  • Smooth bases and relative geometry: If f is smooth, T_f is a vector bundle, and the theorem reads cleanly as a statement about the interaction between the geometry of the fibers and the sheaf E on X.

Generalizations and related results

  • Singular varieties and virtual tangent data: Fulton and others extended the GRR framework to settings with singularities by introducing virtual tangent bundles and refined Chow theories. These refinements maintain the spirit of ch(E) and Td(T_f) while accommodating the failure of smoothness.
  • Connections to Hirzebruch-Riemann-Roch and the index theorem: GRR sits alongside versions like the Hirzebruch-Riemann-Roch theorem in complex geometry and the Atiyah-Singer index theorem in differential geometry, highlighting a deep unity among algebraic, topological, and analytic viewpoints.
  • Derived categories and six-functor formalism: Modern expositions place GRR in the broader language of derived categories and Grothendieck’s six operations, underscoring how pushforwards, pullbacks, tensor products, and duality interact under a common formalism.
  • Computational and applicative reach: The theorem has widespread consequences in enumerative geometry, moduli problems, and the computation of Euler characteristics and Hilbert polynomials, translating geometric questions into manageable algebraic invariants.

Applications

  • Euler characteristics and Hilbert polynomials: For a coherent sheaf E on X, GRR expresses global invariants like χ(X, E) in terms of degrees and intersection numbers on X, after passing through the Chern character and Todd class. This unifies several classical computations into a single framework.
  • Genera and characteristic numbers: The theorem explains how the pushforward of sheaf cohomology reflects the geometry of X and of the morphism f, linking invariants such as the genus in the curve case to higher-dimensional analogues.
  • Moduli problems and intersection theory: In the study of families of varieties and their sheaves, GRR provides a powerful constraint relating fiberwise data to total-space invariants, feeding into calculations of virtual classes and intersection numbers on moduli spaces.

Controversies and debates

As with Grothendieck’s broader program, discussions surrounding GRR often center on the level of abstraction and the balance between generality and concreteness. Key themes include: - Abstraction vs computability: Some mathematicians welcomed the broad, categorical approach that GRR embodies, while others emphasized the desire for more concrete, hands-on formulations in specific geometric settings. - Foundations and frameworks: The use of derived categories, K-theory, and virtual tangent data can be technically demanding. Debates have circulated about the most transparent or practical foundations for working with pushforwards and characteristic classes, especially in the singular or arithmetic cases. - Extent of generality: While GRR has powerful general forms, the precise hypotheses (smoothness of X and Y, properness of f, coefficient fields, base schemes) influence how easily the theorem can be applied. Conversations about extending the scope continue to shape ongoing research in intersection theory and motivic homotopy theory.

See also