nLab
Grothendieck-Riemann-Roch theorem

Idea

The Grothendieck-Riemann-Roch theorem describes (the failure of) the naturality of the behaviour of a Chern character under push forward along proper maps.

It says, in the formulation of (Atiyah-Hirzebruch 61), that the Chern character of the push-forward of some $E$ is the push-forward of the cup product of the Chern-character of $E$ with the Todd class. Hence it says that “Chern-cup-Todd is natural under pushforward” along proper maps, and generlly along K-oriented maps.

If $f:X\to Y$ is a proper map, then there is a commuting diagram

$$\begin{array}{ccc}{K}_0(X)& \stackrel{\mathrm{ch}(-)\cup \mathrm{Td}(X)}{\to }& H_{•}(X,\mathbb{Q})\\ {\downarrow }^{{\int }_f}& & {\downarrow }^{{\int }_f}\\ K_0(Y)& \stackrel{\mathrm{ch}(-)\cup \mathrm{Td}(Y)}{\to }& H_{•}(Y,\mathbb{Q})\end{array},$$\array{ K_0(X) &\stackrel{ch(-)\cup Td(X)}{\to}& H_\bullet(X, \mathbb{Q}) \\ \downarrow^{\mathrlap{\int_{f}}} && \downarrow^{\mathrlap{\int_f}} \\ K_0(Y) &\stackrel{ch(-)\cup Td(Y)}{\to}& H_\bullet(Y,\mathbb{Q}) } \,,

where…

References

The formulation in terms of topological K-theory is due to

• Michael Atiyah, Friedrich Hirzebruch, Vector bundles and homogeneous spaces, Proc. Symp. Pure Math. 3, 7–38 (1961)

For a general survey see

• Wikipedia, Grothendieck-Riemann-Roch theorem

The refinement to differential cohomology, hence differential K-theory, is discussed in section 6.2 of

• Ulrich Bunke, Thomas Schick, Differential K-theory. A survey (arXiv:1011.6663)

Over algebraic stacks/Deligne-Mumford stacks the GRR theorem is discussed in

• Roy Joshua, Riemann-Roch for algebraic stacks (pdf I, pdf II, pdf III)

• Bertrand Toën, Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford, K-Theory 18 (1999), no. 1, 33–76. 1, 23

• Dan Edidin, Riemann-Roch for Deligne-Mumford stacks (arXiv:1205.4742v1)