Chern Character and Riemann--Roch

The Chern character is the fundamental bridge between algebraic K-theory and intersection theory. It transforms the additive but complicated K-group $K_0(X)$ into the more computable Chow ring $CH^*(X) \otimes \mathbb{Q}$ . The Grothendieck--Riemann--Roch theorem describes how the Chern character interacts with proper pushforward.

Chern classes and the Chern character

Definition5.4Chern classes in K-theory

For a smooth variety $X$ over a field, the Chern classes of a vector bundle $\mathcal{E}$ of rank $r$ are elements $c_i(\mathcal{E}) \in CH^i(X)$ for $0 \leq i \leq r$ , defined axiomatically by:

$c_0(\mathcal{E}) = 1$ , $c_i(\mathcal{E}) = 0$ for $i > r$ .
Normalization: For a line bundle $\mathcal{L}$ , $c_1(\mathcal{L}) = [\operatorname{div}(s)]$ for any rational section $s$ .
Whitney sum formula: $c_k(\mathcal{E} \oplus \mathcal{F}) = \sum_{i+j=k} c_i(\mathcal{E}) \cdot c_j(\mathcal{F})$ .
Pullback: $c_i(f^*\mathcal{E}) = f^* c_i(\mathcal{E})$ .

The total Chern class is $c(\mathcal{E}) = 1 + c_1(\mathcal{E}) + c_2(\mathcal{E}) + \cdots \in CH^*(X)$ .

Definition5.5Chern character

The Chern character is the ring homomorphism

$\operatorname{ch}: K_0(X) \to CH^*(X) \otimes \mathbb{Q}$

defined by: if $\mathcal{E}$ has Chern roots $\alpha_1, \ldots, \alpha_r$ (formal variables with $c_k(\mathcal{E}) = e_k(\alpha_1, \ldots, \alpha_r)$ ), then

$\operatorname{ch}(\mathcal{E}) = \sum_{i=1}^r e^{\alpha_i} = r + c_1 + \frac{c_1^2 - 2c_2}{2} + \frac{c_1^3 - 3c_1c_2 + 3c_3}{6} + \cdots$

Explicitly:

$\operatorname{ch}_0(\mathcal{E}) = \operatorname{rk}(\mathcal{E})$
$\operatorname{ch}_1(\mathcal{E}) = c_1(\mathcal{E})$
$\operatorname{ch}_2(\mathcal{E}) = \frac{1}{2}(c_1^2 - 2c_2)$
$\operatorname{ch}_3(\mathcal{E}) = \frac{1}{6}(c_1^3 - 3c_1c_2 + 3c_3)$

The Chern character is a ring homomorphism: $\operatorname{ch}(\mathcal{E} \otimes \mathcal{F}) = \operatorname{ch}(\mathcal{E}) \cdot \operatorname{ch}(\mathcal{F})$ and $\operatorname{ch}(\mathcal{E} \oplus \mathcal{F}) = \operatorname{ch}(\mathcal{E}) + \operatorname{ch}(\mathcal{F})$ .

ExampleChern character of line bundles

For a line bundle $\mathcal{L}$ with $c_1(\mathcal{L}) = D \in CH^1(X)$ :

$\operatorname{ch}(\mathcal{L}) = e^D = 1 + D + \frac{D^2}{2} + \frac{D^3}{6} + \cdots$

On $\mathbb{P}^n$ : $\operatorname{ch}(\mathcal{O}(d)) = e^{dH}$ where $H \in CH^1(\mathbb{P}^n)$ is the hyperplane class. Thus:

$\operatorname{ch}(\mathcal{O}(d)) = 1 + dH + \frac{d^2}{2}H^2 + \cdots + \frac{d^n}{n!}H^n$

since $H^{n+1} = 0$ in $CH^*(\mathbb{P}^n)$ .

The Grothendieck--Riemann--Roch theorem

Theorem5.1Grothendieck--Riemann--Roch (GRR)

Let $f: X \to Y$ be a proper morphism of smooth varieties over a field. For any $\alpha \in K_0(X)$ :

$\operatorname{ch}(f_!\alpha) \cdot \operatorname{td}(T_Y) = f_*(\operatorname{ch}(\alpha) \cdot \operatorname{td}(T_X))$

in $CH^*(Y) \otimes \mathbb{Q}$ , where:

$f_!: K_0(X) \to K_0(Y)$ is $f_!(\mathcal{E}) = \sum_i (-1)^i [R^i f_* \mathcal{E}]$ (the K-theoretic pushforward).
$\operatorname{td}(T_X)$ is the Todd class of the tangent bundle $T_X$ : $\operatorname{td}(\mathcal{E}) = \prod_{i=1}^r \frac{\alpha_i}{1 - e^{-\alpha_i}} = 1 + \frac{c_1}{2} + \frac{c_1^2 + c_2}{12} + \frac{c_1 c_2}{24} + \cdots$
$f_*: CH^*(X) \to CH^*(Y)$ is the proper pushforward on Chow groups.

Equivalently, setting $\operatorname{td}(f) = \operatorname{td}(T_X) / f^*\operatorname{td}(T_Y)$ (the relative Todd class):

$\operatorname{ch}(f_!\alpha) = f_*(\operatorname{ch}(\alpha) \cdot \operatorname{td}(T_f))$

where $T_f$ is the relative tangent bundle.

Special cases

ExampleHirzebruch--Riemann--Roch

When $Y = \operatorname{Spec} k$ (a point) and $X$ is a smooth projective variety of dimension $d$ :

$\chi(X, \mathcal{E}) = \sum_{i=0}^d (-1)^i \dim H^i(X, \mathcal{E}) = \int_X \operatorname{ch}(\mathcal{E}) \cdot \operatorname{td}(T_X)$

where $\int_X$ means taking the degree- $d$ component and its degree.

For a curve $C$ of genus $g$ and line bundle $\mathcal{L}$ of degree $d$ :

$\chi(C, \mathcal{L}) = h^0(\mathcal{L}) - h^1(\mathcal{L}) = d + 1 - g$

recovering the Riemann--Roch theorem: $\operatorname{ch}(\mathcal{L}) = 1 + d[\text{pt}]$ , $\operatorname{td}(T_C) = 1 + (1-g)[\text{pt}]$ , and $\int_C (1 + d[\text{pt}])(1 + (1-g)[\text{pt}]) = d + 1 - g$ .

For a surface $S$ and line bundle $\mathcal{L}$ with $c_1(\mathcal{L}) = D$ :

$\chi(S, \mathcal{L}) = \frac{D \cdot (D - K_S)}{2} + \chi(\mathcal{O}_S)$

where $K_S$ is the canonical divisor and $\chi(\mathcal{O}_S) = \frac{c_1^2 + c_2}{12}$ (Noether's formula, with $c_i = c_i(T_S)$ ).

ExampleGRR for a fibration

For a smooth fibration $f: X \to Y$ with fiber $F$ of dimension $d$ , applied to $\alpha = [\mathcal{O}_X]$ :

$\operatorname{ch}(f_![\mathcal{O}_X]) = f_*(\operatorname{td}(T_f))$

The left side is $\operatorname{ch}(\sum (-1)^i R^i f_* \mathcal{O}_X)$ and the right side involves the Todd class of the relative tangent bundle. For a family of curves ( $d = 1$ ):

$\operatorname{ch}(f_* \mathcal{O}_X - R^1 f_* \mathcal{O}_X) = f_*\left(1 + \frac{c_1(T_f)}{2} + \frac{c_1(T_f)^2}{12}\right)$

The degree-0 part gives $\operatorname{rk}(f_* \mathcal{O}_X) - \operatorname{rk}(R^1 f_* \mathcal{O}_X) = 1 - g$ (fibers have genus $g$ ). The degree-1 part gives $c_1(f_* \mathcal{O}_X) - c_1(R^1 f_* \mathcal{O}_X) = \frac{1}{2} f_* c_1(T_f) = \frac{1}{12}\kappa_1$ where $\kappa_1$ is the first Mumford class.

The rational isomorphism

RemarkChern character is a rational isomorphism

For a smooth variety $X$ over a field, the Chern character induces an isomorphism

$\operatorname{ch}: K_0(X) \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\;\sim\;} CH^*(X) \otimes_{\mathbb{Z}} \mathbb{Q}.$

This means that rationally, K-theory and Chow theory carry the same information. The integral difference is measured by:

Torsion in $K_0(X)$ : always maps to zero in $CH^* \otimes \mathbb{Q}$ .
Adams operations: $\psi^k$ acts on $\operatorname{gr}^p K_0(X) \otimes \mathbb{Q}$ by multiplication by $k^p$ . The eigenspaces of $\psi^k$ on $K_0(X) \otimes \mathbb{Q}$ correspond to $CH^p(X) \otimes \mathbb{Q}$ .

The integral refinement $K_0(X) \to CH^*(X)$ (without $\otimes \mathbb{Q}$ ) is given by the Chern class $c_n$ (not $\operatorname{ch}$ ) and is in general neither injective nor surjective.