5.4 Grothendieck-Riemann-Roch theorem

The Grothendieck-Riemann-Roch theorem is a game-changer in algebraic geometry. It links sheaf pushforwards to Chern characters and Todd classes, generalizing earlier Riemann-Roch results for curves and complex manifolds.

This powerful tool helps calculate important invariants like Euler characteristics. By connecting K-theory and Chow rings, it bridges vector bundles and intersection theory, opening new avenues for studying algebraic varieties and their properties.

Grothendieck-Riemann-Roch Theorem

Statement of the Theorem

• The Grothendieck-Riemann-Roch theorem generalizes the classical Riemann-Roch theorem for curves and the Hirzebruch-Riemann-Roch theorem for complex manifolds
• Let $f: X \to Y$ be a proper morphism between smooth quasi-projective varieties over a field
  • The theorem states that for any coherent sheaf $E$ on $X$, the following equality holds in the Chow ring of $Y$: $ch(f_E) = f_(ch(E) \cdot td(Tf))$
  • $ch$ denotes the Chern character, $td$ denotes the Todd class, and $Tf$ is the relative tangent bundle of $f$
• The theorem relates the pushforward of a coherent sheaf $E$ on $X$ to the pushforward of the product of its Chern character and the Todd class of the relative tangent bundle
  • The Chern character and Todd class encode important information about the sheaf and the morphism, respectively
• The theorem holds more generally for proper morphisms between smooth schemes over a field, with appropriate modifications to the statement (algebraic spaces, Deligne-Mumford stacks)

Generalization and Unification

• The Grothendieck-Riemann-Roch theorem is a fundamental result in algebraic geometry that generalizes and unifies various previous Riemann-Roch type theorems
  • It extends the classical Riemann-Roch theorem for curves, which relates the degree of a divisor to the dimension of the space of global sections of the associated line bundle
  • It generalizes the Hirzebruch-Riemann-Roch theorem for complex manifolds, which expresses the Euler characteristic of a coherent sheaf in terms of characteristic classes
• The formulation of the theorem in terms of K-theory and Chow rings highlights the importance of these algebraic structures in the study of schemes and their vector bundles
  • K-theory captures information about vector bundles and coherent sheaves, while Chow rings describe the intersection theory of algebraic cycles
• The theorem provides a powerful tool for computing invariants of coherent sheaves, such as Euler characteristics and Hilbert polynomials, in terms of characteristic classes (Chern classes, Todd classes)

Chern Characters and Todd Classes

Chern Character

• The Chern character is a ring homomorphism from the K-theory of a scheme to its Chow ring, which assigns to each vector bundle a formal sum of its Chern classes
  • For a line bundle $L$, the Chern character is given by $ch(L) = \exp(c_1(L))$, where $c_1(L)$ is the first Chern class of $L$
  • The Chern character is additive on short exact sequences: if $0 \to E' \to E \to E'' \to 0$ is a short exact sequence of vector bundles, then $ch(E) = ch(E') + ch(E'')$
  • The Chern character is multiplicative on tensor products of vector bundles: $ch(E \otimes F) = ch(E) \cdot ch(F)$ for vector bundles $E$ and $F$
• The Chern character can be defined axiomatically by its properties, or explicitly in terms of Chern classes using the formal power series $\exp(x) = \sum_{n=0}^\infty \frac{x^n}{n!}$

Todd Class

• The Todd class is a characteristic class associated with a vector bundle, defined in terms of its Chern classes
  • For a line bundle $L$, the Todd class is given by $td(L) = c_1(L)/(1-\exp(-c_1(L)))$
  • The Todd class satisfies the multiplicative property: $td(E \oplus F) = td(E) \cdot td(F)$ for vector bundles $E$ and $F$
• In the Grothendieck-Riemann-Roch theorem, the Todd class of the relative tangent bundle $Tf$ measures the difference between the tangent bundles of $X$ and $Y$
  • The relative tangent bundle fits into the short exact sequence $0 \to f^*T_Y \to T_X \to Tf \to 0$, where $T_X$ and $T_Y$ are the tangent bundles of $X$ and $Y$, respectively
• The Todd class can be expressed as a power series in the Chern classes, with rational coefficients (Bernoulli numbers)

Euler Characteristics Calculation

Definition of Euler Characteristic

• The Euler characteristic of a coherent sheaf $E$ on a proper scheme $X$ over a field is defined as $\chi(X, E) = \sum_i (-1)^i \dim_k H^i(X, E)$
  • $H^i(X, E)$ denotes the $i$-th cohomology group of the sheaf $E$ on $X$
  • The alternating sum of the dimensions of the cohomology groups gives the Euler characteristic
• The Euler characteristic is an important invariant that captures topological and algebraic information about the sheaf and the underlying scheme

Application of Grothendieck-Riemann-Roch

• The Grothendieck-Riemann-Roch theorem can be used to compute Euler characteristics by comparing the pushforward of a sheaf to the pushforward of its Chern character
  • In the case of a morphism $f: X \to \operatorname{Spec}(k)$ from a smooth projective variety to a point, the theorem simplifies to $\chi(X, E) = \deg(ch(E) \cdot td(TX))$
  • $\deg$ denotes the degree map from the Chow ring of $X$ to the ground field $k$, which sends a zero-cycle to its degree
• By expressing the Chern character and Todd class in terms of Chern classes, one can compute the Euler characteristic of a sheaf using the intersection theory of the variety $X$
  • The computation involves expanding the product $ch(E) \cdot td(TX)$ in the Chow ring and applying the degree map
  • The resulting formula expresses the Euler characteristic in terms of the degrees of certain cycle classes on $X$ (intersection numbers)

Significance in Algebraic Geometry

Impact on the Field

• Grothendieck's proof of the theorem, based on the construction of the Grothendieck group of coherent sheaves and the Chern character, introduced new techniques and ideas that have had a profound impact on the development of algebraic geometry
  • The use of K-theory to study vector bundles and coherent sheaves has become a fundamental tool in algebraic geometry
  • The Grothendieck group construction allows for the definition of generalized Euler characteristics and the formulation of Riemann-Roch type theorems in various settings
• The theorem has led to further generalizations and analogues, such as the Atiyah-Singer index theorem in differential geometry and the Baum-Fulton-MacPherson theorem in intersection theory

Applications and Consequences

• The Grothendieck-Riemann-Roch theorem has numerous applications in the theory of moduli spaces, the study of algebraic cycles, and the computation of intersection numbers
  • It provides a way to compute the Euler characteristics of families of varieties parametrized by a moduli space, which is important in the study of enumerative problems
  • It relates the intersection theory of a variety to the K-theory of its coherent sheaves, allowing for the computation of Chow rings and intersection numbers using vector bundles
• The theorem has consequences in the theory of characteristic classes, as it provides relations between different types of characteristic classes (Chern classes, Todd classes, Chern characters)
  • These relations have been used to prove vanishing theorems, study the topology of algebraic varieties, and compute invariants of vector bundles
• The Grothendieck-Riemann-Roch theorem is a cornerstone of modern algebraic geometry and has inspired a wide range of research in related fields, such as arithmetic geometry, topological K-theory, and algebraic topology