Grothendieck--Riemann--Roch Theorem

The Grothendieck--Riemann--Roch (GRR) theorem is the cornerstone of the interaction between K-theory and intersection theory. It describes the precise failure of the Chern character to commute with proper pushforward, the correction being the Todd class. This result subsumes the classical Riemann--Roch and Hirzebruch--Riemann--Roch theorems.

Statement

Theorem5.2Grothendieck--Riemann--Roch

Let $f: X \to Y$ be a proper morphism of smooth quasi-projective varieties over a field $k$ . For every element $\alpha \in K_0(X)$ :

$\operatorname{ch}(Rf_*(\alpha)) = f_*\left(\operatorname{ch}(\alpha) \cdot \operatorname{td}(T_f)\right) \quad \text{in } CH^*(Y) \otimes \mathbb{Q}$

where:

$Rf_*: K_0(X) \to K_0(Y)$ is the derived pushforward: $Rf_*([\mathcal{E}]) = \sum_{i \geq 0} (-1)^i [R^i f_* \mathcal{E}]$ .
$\operatorname{ch}: K_0 \to CH^* \otimes \mathbb{Q}$ is the Chern character.
$\operatorname{td}(T_f) = \operatorname{td}(T_X) / f^*\operatorname{td}(T_Y)$ is the Todd class of the relative tangent sheaf.
$f_*: CH^*(X) \to CH^*(Y)$ is the proper pushforward on Chow groups.

The theorem says: the Chern character does not commute with pushforward, but the discrepancy is measured by the Todd class.

Proof sketch

Proof

The proof reduces to two special cases by factoring $f$ as a closed immersion followed by a projection.

Step 1: Factor $f$ . By Nagata's compactification and Chow's lemma, we may assume $X$ and $Y$ are projective. Embed $X \hookrightarrow \mathbb{P}^N_Y$ for some $N$ , so $f$ factors as

$X \xrightarrow{i} \mathbb{P}^N_Y \xrightarrow{p} Y$

where $i$ is a closed immersion and $p$ is the projection. It suffices to prove GRR for $i$ and $p$ separately (since both sides are multiplicative under composition).

Step 2: GRR for projective bundle projection $p: \mathbb{P}(\mathcal{E}) \to Y$ . This follows from the projective bundle theorem. The pushforward $p_*: K_0(\mathbb{P}(\mathcal{E})) \to K_0(Y)$ is computed using the Koszul resolution and the formula for $R^i p_* \mathcal{O}(j)$ . The Chern character computation reduces to the identity:

$\sum_{j=0}^{r-1} \frac{e^{jt}}{(1 - e^{-t})^r} \cdot \operatorname{td}_r = \text{(appropriate sum)}$

which is a formal identity in the Todd class.

Step 3: GRR for a closed immersion $i: X \hookrightarrow Y$ . This is the substantial case. The pushforward $i_*: K_0(X) \to K_0(Y)$ sends $[\mathcal{E}]$ to $\sum (-1)^j [\mathcal{F}_j]$ where $\mathcal{F}_\bullet \to i_*\mathcal{E} \to 0$ is a locally free resolution on $Y$ .

The key identity to prove is:

$\operatorname{ch}(i_*\alpha) = i_*\left(\operatorname{ch}(\alpha) \cdot \operatorname{td}(\mathcal{N}_{X/Y})^{-1}\right)$

where $\mathcal{N}_{X/Y}$ is the normal bundle, using $\operatorname{td}(T_f) = \operatorname{td}(T_X)/i^*\operatorname{td}(T_Y) = \operatorname{td}(\mathcal{N}_{X/Y})^{-1}$ .

For a regular embedding of codimension $c$ , with normal bundle $\mathcal{N}$ of rank $c$ , the Koszul complex $\Lambda^\bullet \mathcal{N}^\vee \to \mathcal{O}_X \to 0$ gives:

$[i_*\mathcal{O}_X] = \sum_{j=0}^c (-1)^j [\Lambda^j \mathcal{N}^\vee]$

and one computes $\operatorname{ch}(\sum (-1)^j \Lambda^j \mathcal{N}^\vee) = \prod_{i=1}^c (1 - e^{-\alpha_i})$ where $\alpha_i$ are the Chern roots of $\mathcal{N}$ .

Step 4: The self-intersection formula. The relation $i^* i_*(1) = c_c(\mathcal{N}^\vee) = (-1)^c c_c(\mathcal{N})$ (the Euler class) is used together with the deformation to the normal cone technique to handle the general case.

Step 5: Combine. Since GRR holds for projections and closed immersions, and $f = p \circ i$ , the theorem follows for all proper morphisms. $\square$

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Examples

ExampleNoether's formula for surfaces

For a smooth projective surface $S$ over an algebraically closed field, applying GRR to $f: S \to \operatorname{Spec} k$ with $\alpha = [\mathcal{O}_S]$ :

$\chi(\mathcal{O}_S) = \int_S \operatorname{td}(T_S) = \int_S \left(1 + \frac{c_1}{2} + \frac{c_1^2 + c_2}{12}\right) = \frac{c_1^2(T_S) + c_2(T_S)}{12}$

Since $c_1(T_S) = -K_S$ and $c_2(T_S) = e(S)$ (topological Euler characteristic):

$\chi(\mathcal{O}_S) = \frac{K_S^2 + e(S)}{12}$

This is Noether's formula. For $S = \mathbb{P}^2$ : $K_{\mathbb{P}^2} = \mathcal{O}(-3)$ , $K^2 = 9$ , $e = 3$ , $\chi(\mathcal{O}) = 12/12 = 1$ .

ExampleDirect image of a line bundle on a curve

For a finite morphism $f: C \to D$ of smooth projective curves of degree $d$ , and a line bundle $\mathcal{L}$ of degree $n$ on $C$ :

$\operatorname{ch}(f_*\mathcal{L}) = f_*(\operatorname{ch}(\mathcal{L}) \cdot \operatorname{td}(T_f))$

The relative tangent bundle has $\operatorname{td}(T_f) = 1 + \frac{1}{2}c_1(T_f)$ . By Riemann--Hurwitz, $c_1(T_f) = c_1(\Omega_C^\vee) - f^*c_1(\Omega_D^\vee) = (2-2g_C) - d(2-2g_D)$ as a degree on $C$ .

The degree-0 component gives $\operatorname{rk}(f_*\mathcal{L}) = f_*(1) = d$ . The degree-1 component gives $c_1(f_*\mathcal{L}) = f_*(n + \frac{1}{2}c_1(T_f)) = n + d(1-g_D) - (1-g_C)$ . Combining: $\deg(f_*\mathcal{L}) = n + d(1-g_D) - (1-g_C)$ .

RemarkK-theoretic vs cohomological formulation

GRR can be stated purely in K-theory, without the Chern character. For a proper morphism $f: X \to Y$ , the diagram

$\begin{array}{ccc} K_0(X) & \xrightarrow{\operatorname{ch}(-) \cdot \operatorname{td}(T_X)} & CH^*(X) \otimes \mathbb{Q} \\ \downarrow^{Rf_*} & & \downarrow^{f_*} \\ K_0(Y) & \xrightarrow{\operatorname{ch}(-) \cdot \operatorname{td}(T_Y)} & CH^*(Y) \otimes \mathbb{Q} \end{array}$

commutes. Equivalently, the modified Chern character $\tau = \operatorname{ch}(-) \cdot \operatorname{td}(T)$ is a natural transformation from the K-theory functor to the Chow functor (both considered with proper pushforward).