Contents

Idea

The Chern character of a generalized (Eilenberg-Steenrod) cohomology theory is a canonical morphism from the generalized cohomology to ordinary (Eilenberg Mac-Lane) real cohomology. When thought of in the refinement to differential cohomology and thinking of a cocycle in differential cohomology as a generalization of a connection on a bundle, the Chern-character is the map that sends a generalized connection to its curvature characteristic form.

More in detail, for every generalized (Eilenberg-Steenrod) cohomology theory given by a spectrum $E$, there is a canonical natural isomorphism from the rationalized $E$-cohomology to ordinary (Eilenberg-MacLane) cohomology with coefficients in the rationalized homotopy groups of $E$:

So rationally, every generalized (Eilenberg-Steenrod) cohomology is ordinary (Eilenberg-MacLane) cohomology. Hence every generalized (ES-)cohomology theory has a canonical morphism to ordinary real cohomology

In the case that $E=\mathrm{KU}$ is the K-theory spectrum, this morphism is classically known as the Chern character. Generalizing from this example, the term “Chern character” is sometimes used also for the general case.

There are analogues in algebraic geometry (e.g. a Chern character between the Chow groups and the algebraic K-theory) and noncommutative geometry (Chern-Connes character) where the role of usual cohomology is taken by some variant of cyclic cohomology.

For vector bundles and K-theory

The classical theory of the Chern character applies to the spectrum of complex K-theory, $E=\mathrm{KU}$. In this case, the Chern character is made up from Chern classes: each characteristic class is by Chern-Weil theory in image of certain element in the Weil algebra via taking the class of evaluation at the curvature operator for some choice of a connection. Consider the symmetric functions in $n$ variables $t_1,\dots ,t_n$ and let the Chern classes of a complex vector bundle $\xi$ (representing a complex K-theory class) are $c_1,\dots ,c_n$. Define the formal power series

Then $\mathrm{ch}(\chi )=\varphi (c_1,\dots ,c_n)$.

Let us describe this a bit differently. The cocycle $H^0(X,\mathrm{KU})$ may be represented by a complex vector bundle, and the image of this cocycle under the Chern-character is the class in even-graded real cohomology that is represented (under the deRham theorem isomorphism of deRham cohomology with real cohomology) by the even graded closed differential form

where

• $\nabla$ is any chosen connection on the vector bundle;

• $F=F_{\nabla }\in {\Omega }^2(X,\mathrm{End}(V))$ is the curvature of this connection;

• $k_j\in ℝ$ are normalization constants, $k_j=\frac{1}{j!}{\left(\frac{1}{2\pi i}\right)}^j$;

• the trace of the wedge products produces the curvature characteristic forms.

The Chern character applied to the Whitney sum of two vector bundles is a sum of the Chern characters for the two: $\mathrm{ch}(\xi \oplus \eta )=\mathrm{ch}(\chi )+\mathrm{ch}(\eta )$ and it is multiplicative under the tensor product of vector bundles: $\mathrm{ch}(\xi \otimes \eta )=\mathrm{ch}(\chi )\mathrm{ch}(\eta )$. Therefore we get a ring homomorphism.

General theory

The fundamental cocycle

The isomorphism

that defines the Chern-character map is induced by a canonical cocycle on the spectrum $E$ that is called the fundamental cocycle.

This is described for instance in section 4.8, page 47 of Hopkins-Singer Quadratic Functions in Geometry, Topology,and M-Theory.

Properties

Push-forward

The behaviour of the Chern-character under fiber integration in generalized cohomology along proper maps is described by the Grothendieck-Riemann-Roch theorem.

References

The universal Chern character for generalized (Eilenberg-Steenrod) cohomology theory is discussed in section 4.8, page 47 of

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology,and M-Theory, (math.AT/0211216).

A characterization of Chern-character maps for K-theory is in

• Goncalo Tabuada, A universal characterization of the Chern character maps, arXiv/1002.3276)

A discussion of Chern characters in terms of free loop space objects in derived geometry is in

• Bertrand Toën, Gabriele Vezzosi, A note on Chern character, loop spaces and derived algebraic geometry (arXiv:0804.1274)

which conjectures a construction that is fully developed in

• Bertrand Toën, Gabriele Vezzosi, Caractères de Chern, traces équivariantes et géométrie algébrique dérivée (arXiv:0903.3292)

See also

• Marc Hoyois, Chern character and derived algebraic geometry (2009) (pdf)