Traditionally, in the strict sense of the term, the Chern character is a universal characteristic class of vector bundles or equivalently of their topological K-theory classes, which is a rational combination of all Chern classes.
This is a special case of the following more general construction (Hopkins-Singer 02, section 4.8):
for a spectrum representing a generalized (Eilenberg-Steenrod) cohomology theory there is a canonical localization map
In the case that KU this reproduces the traditional Chern character. (In which case this is a map from a complex oriented cohomology theory of chromatic level 1 to chromatic level 0. More generally one can also consider higher chromatic Chern characters that take values not in ordinary cohomology but in some cohomology theory of higher chromatic level. See at higher chromatic Chern character for more on this.)
The Chern character may be used to define differential cohomology refinements of the cohomology theory by choosing a differential form-model for (Hopkins-Singer 02, see also at differential function complex). In that case is the real cohomology class associated to a chern character differential form via the de Rham theorem. Here has the interpretation of being the curvature forms of the differential cohomology cocycles thought of as ∞-connections.
This may be turned around (Bunke-Nikolaus-Völkl 13, prop. .3.5): given any refinement of in a tangent cohesive (∞,1)-topos , then it is induced from homotopy pullback of its de Rham coefficients along a Chern character map
where is the shape modality and the Maurer-Cartan form of . This reproduces the above definition for ordinary differential form models, see at differential cohomology diagram – Hopkins-Singer coefficients.
But more generally, given for instance a K(n)-localization then any choice of cohesive refinement of (i.e. lift through the unit of the shape modality ) which is in the kernel of yields a generalized differential cohomology theory whose intrinsic Chern-character is the -localization. See at differential cohomology diagram – Chern character and differential fracture.
In the context of algebraic K-theory Chern characters appear at Beilinson regulators. There are analogues in algebraic geometry (e.g. a Chern character between Chow groups and algebraic K-theory) and in noncommutative geometry (Chern-Connes character) where the role of usual cohomology is taken by some variant of cyclic cohomology.
The classical theory of the Chern character applies to the spectrum of complex K-theory, . In this case, the Chern character is made up from Chern classes: each characteristic class is by Chern-Weil theory in the image of a 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 variables and let the Chern classes of a complex vector bundle (representing a complex K-theory class) be . Define the formal power series
Let us describe this a bit differently. The cocycle 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
is any chosen connection on the vector bundle;
is the curvature of this connection;
are normalization constants, ;
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: and it is multiplicative under the tensor product of vector bundles: . Therefore we get a ring homomorphism.
that defines the Chern-character map is induced by a canonical cocycle on the spectrum 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.
For more on this see at differential cohomology diagram.
Generalizing in another direction, generalized Chern characters are given by passage to derived loop spaces and their cyclic homology or, more generally, topological cyclic homology (Toen-Vezzosi 08, Hoyois-Scherotzke-Sibillia 15).
for algebraic K-theory:
The universal Chern character for generalized (Eilenberg-Steenrod) cohomology theory is discussed in section 4.8, page 47 of
A characterization of Chern-character maps for K-theory is in
which conjectures a construction that is fully developed in